C, 358 bytes

(There are only three lines of code, but I've broken up line 3 for legibility)

#define x putchar
#define y random()
c,n;main(){char*s[26]={"QS","HNV","FVX","EFSX","DRW","CDGR","FHTV","BGJY","KOU","HKNU",
"IJLM","KO","KN","BJM","ILP","OO","AW","EFT","ADWZ","GRY","IJY","BCG","ESQ","CDZ","HTU",
"SX"};while((c=getchar())>0){if(y%10>0&&x(c))continue;if(isalpha(c)&&y%3<1){n=(c&31)-1;
x(s[n][y%strlen(s[n])]|(c&32));continue;}if (y&1)x(x(c));}}

The array of strings at the beginning lists the possible adjacent keys for each letter of the alphabet. I had to double up the "O" (adjacent to "P") to avoid calculating random()%1 when selecting a shifted character.

Test run:

$ echo "This is some correct text. It is too correct. Please un-correctify it." |./a.out
This is some  cofrect teext. It is too correct.. Plleaase un-correctify it..

Update:

Here's an expanded and commented version of the same source code:

#include <stdio.h>
#include <string.h>
/* ^^ These should be included, but the code compiles without them */

int c,n;

void main() {

  /* Adjacent keys for each letter of the alphabet (A, B, C, ..., Z): */
  char *s[26] = { "QS","HNV","FVX","EFSX","DRW","CDGR","FHTV","BGJY","KOU","HKNU",
                  "IJLM","KO","KN","BJM","ILP","OO","AW","EFT","ADWZ","GRY","IJY",
                  "BCG","ESQ","CDZ","HTU","SX" };

  /* Fetch input until null character or EOF reached */
  while ((c=getchar())>0) {

    /* For 90% of the time, just echo the character unchanged */
    if (random()%10>0 && putchar(c)) continue;

    /* If it's a letter of the alphabet, shift with 33% probability */
    if (isalpha(c) && random()%3<1) {
      /* Calculate offset to adjacent keys data */
      n=(c&31)-1;
      /* Choose a random adjacent key, and change to lower case if needed */
      putchar(s[n][random()%strlen(s[n])]|(c&32));
      continue;
    }

    /* If we reach this point, either we didn't fetch an alphabet character, or   */
    /* we did but chose not to shift it. Either way, we now need to either repeat */
    /* the character or delete it, with 50% probability for each. */

    /* This repeats the character by printing the return value from putchar() */
    if (random()&1) putchar(putchar(c));

    /* To delete the character, we don't have to do anything. */
  }
}

Ruby, 183

s='.qwertyuiop.asdfghjkl...zxcvbnm.'
gets.chars{|c|c=~/\w/&&(s[c]?s: s.upcase)=~/((\w).{9})?((\w)|.)#{c}((\w)|.)(.{9}(\w))?/
putc rand<0.9?c:[c*2,'',*[*$2,*$4,*$6,*$8].sample].sample}

Python, 249

from random import*
w=choice
o=ord
print"".join(w([z]*9+[w(["",z*2]+[chr(o(w("SQ VNH XVF SFEX WRD DGRC FHTV GJYB UOK HKUN JLIM KO NK BMJ IPL O WA ETF ADWZ RYG YIJ CBG QES ZCD TUH XS".split()[o(z)&31-1]))|o(z)&32)]*z.isalpha())])for z in raw_input())

Very simple look-up technique, for a moment I thought it might be cheaper to encode the keyboard as a undirected graph but the overhead for creating such a type in Python proved prohibitive. Since Python's random functions have too-descriptive names I use choice() exclusively, renaming it to w. The 10% chance of error is handled by w([z]*9+[...]) where the nine copies of an un-typoed character are in a list with one typo.

_{-16 characters--thanks grc.}

C#, 320 bytes (360 bytes with program wrapping)

Includes support for "shifted" upper-case letters.

As a function (320 bytes):

string T(string s){
    string t="",z="      ",k=z+z+"qwertyuiop asdfghjkl   zxcvbnm";
    k+=k.ToUpper()+z+z;
    int[]m={-11,-1,1,11};
    var r=new System.Random();
    foreach(var c in s){
        if(r.Next(10)>0)
            t+=c;
        else{
            int i=r.Next(k.IndexOf(c)>0?3:2);
            if(i>1){
                while(' '==k[i=k.IndexOf(c)+m[r.Next(4)]]);
                t+=k[i];
            }
            else if(i>0)
                t=(t+c)+c;
        }
    }
    return t;
}

As a program that reads a line of text (360 bytes):

using System;
class P{
    static void Main(){
        string t="",z="      ",k=z+z+"qwertyuiop asdfghjkl   zxcvbnm";
        k+=k.ToUpper()+z+z;
        int[]m={-11,-1,1,11};
        var r=new Random();
        foreach(var c in Console.ReadLine()){
            if(r.Next(10)>0)
                t+=c;
            else{
                int i=r.Next(k.IndexOf(c)>0?3:2);
                if(i>1){
                    while(' '==k[i=k.IndexOf(c)+m[r.Next(4)]]);
                    t+=k[i];
                }
                else if(i>0)
                    t=(t+c)+c;
            }
        }
        Console.Write(t);
    }
}

Input output sample:

This is some correct text. It is too correct. Please un-correctify it.
This  is some corrrect texy. Ut is too correct. Pease un-coorrectify it.

TYPO RAGE CAPS TEXT!
TYPPORAGE CAOS TEX!

PHP, function with 368 bytes

Here is my attempt.

It's some "frankencode", but it kinda works.

function _($m){$q=array(array('qwertyuiop','asdfghjkl',' zxcvbnm'),'!.-,;?+/');$r='mt_rand';foreach(str_split($m)as$k=>$c)if(!$r(0,9)&&$c!=' ')foreach($q[0]as$x=>$y)if(($z=strpos($y,$c))!==!1){switch($t=$r(-3,2-($z>>3)-($x>>1))){case 2:++$z;break;case 1:++$x;break;case -1:--$x;break;case -2:--$z;break;case -3:$z=8;break;}$m[$k]=$t?$q[0][$x][$z]:$q[1][$z];}return$m;}

A more "readable" code:

function _($m)
{
    $q=array(array('qwertyuiop','asdfghjkl',' zxcvbnm'),'!.-,;?+/');
    $r='mt_rand';
    foreach(str_split($m)as$k=>$c)
        if(!$r(0,9)&&$c!=' ')
            foreach($q[0]as$x=>$y)
                if(($z=strpos($y,$c))!==!1)
                {
                    switch($t=$r(-3,2-($z>>3)-($x>>1)))
                    {
                        case 2:
                            ++$z;break;
                        case 1:
                            ++$x;break;
                        case -1:
                            --$x;break;
                        case -2:
                            --$z;break;
                        case -3:
                            $z=8;break;
                    }
                    $m[$k]=$t?$q[0][$x][$z]:$q[1][$z];
                }
    return$m;
}

The only difference between the 2 codes is that one has tons of tabs and newlines.

It doesn't produce the same exact type of "incorrectness", but it either deletes or replaces a char using the said conditions.

You can try it at http://writecodeonline.com/php/.

Copy and paste this code:

function _($m){$q=array(array('qwertyuiop','asdfghjkl',' zxcvbnm'),'!.-,;?+/');$r='mt_rand';foreach(str_split($m)as$k=>$c)if(!$r(0,9)&&$c!=' ')foreach($q[0]as$x=>$y)if(($z=strpos($y,$c))!==!1){switch($t=$r(-3,2-($z>>3)-($x>>1))){case 2:++$z;break;case 1:++$x;break;case -1:--$x;break;case -2:--$z;break;case -3:$z=8;break;}$m[$k]=$t?$q[0][$x][$z]:$q[1][$z];}return$m;}
echo _('This is some correct text. It is too correct. Please un-correctify it.');

After testing, please, tell me if it is a valid answer.

C#, 581 bytes

using System;class B{static void Main(){var z=Console.ReadLine();var r=new Random();foreach(char C in z){String o;if(r.Next(10)==0){int w=r.Next(3);o=w==0?C+""+C:w==1?"":f(C,r);}else{o=C+"";}Console.Write(o);}Console.ReadLine();}static string f(char C,Random r){string[]k={"00000000000","0qwertyuiop0","0asdfghjkl0","00zxcvbnm0","000000000"};char L=char.ToLower(C);char h='0';for(int i=1;i<4;i++){var d=k[i].IndexOf(L);if(d!=-1){while(h=='0'){int n=r.Next(4);h=n==0?k[i][d-1]:n==1?k[i][d+1]:n==2?k[i-1][d]:k[i+1][d];}h=char.IsUpper(C)?char.ToUpper(h):h;return h+"";}}return C+"";}}

and in a more readable format:

using System;

class A
{
    static void Main()
    {
        var z = Console.ReadLine();
        var r = new Random();

        foreach (char C in z)
        {
            String o;

            if (r.Next(10) == 0)
            {
                int w = r.Next(3);
                o = w == 0 ? C + "" + C :
                    w == 1 ? "" :
                             f(C, r);
            }
            else
            {
                o = C + "";
            }

            Console.Write(o);
        }
    }

    static string f(char C, Random r)
    {
        string[] k = {
                            "00000000000", 
                            "0qwertyuiop0", 
                            "0asdfghjkl0", 
                            "00zxcvbnm0", 
                            "000000000"};  
        char L = char.ToLower(C);
        char h = '0';

        for (int i = 1; i < 4; i++)
        {
            var d = k[i].IndexOf(L);

            if (d != -1)
            {
                while (h == '0')
                {
                    int n = r.Next(4);

                    h = n == 0 ? k[i][d - 1] :
                        n == 1 ? k[i][d + 1] :
                        n == 2 ? k[i - 1][d] :
                                 k[i + 1][d];
                }
                h = char.IsUpper(C) ? char.ToUpper(h) : h;
                return h + "";
            }
        }
        return C + "";
    }
}

Perl, 278 239

sub i{int rand$_[0]};s#.#i(10)?$&:($n=i$&=~/\pL/?3:2)?$n-1?($_=substr+($_=qw/SQ VNH XVF SFEX WRD DGRC FHTV GJYB UOK HKUN JLIM KO NK BMJ IPL O WA ETF ADWZ RYG YIJ CBG QES ZCD TUH XS/[ord(uc$&)-65]),i(length),1)&&($& gt'Z')?lc:$_:$&x2:''#ge

Run with -p, then 238+1=239 bytes. E.g.:

perl -p typos.pl
This is some correct text. It is too correct. Please un-correctify it.
This is  some correct trxt. It is too correct. Pleaseun-correctify it.
I wrote some text, but it looks too professional.
I wrote some ret, htit looks too professiilnal.
I want to make it look like I was really tired when I wrote it.
I want  to mqke it look like I was reaally tired when I wroteit.
I need you to insert some typos.
I eed youu t inseert sooke typos.

Un-golfed, more or less:

sub i{int rand$_[0]};
s#.#
i(10)
    ?$&
    :($n=i$&=~/\pL/?3:2)
        ?$n-1
            ?
                (
                    $_=substr +($_=qw/SQ VNH XVF SFEX WRD DGRC FHTV GJYB UOK 
                        HKUN JLIM KO NK BMJ IPL O WA ETF ADWZ RYG 
                        YIJ CBG QES ZCD TUH XS/[ord(uc$&)-65])
                    ,i(length),1
                )
                &&
                ($& gt'Z')
                    ?lc
                    :$_
            :$&x2
        :''
#ge;

GolfScript, 120 characters

{10{rand}:R~!{[{.}{;}{Z\?[.(.10-@).10+]{Z=}%' '-.,R=}]'QWERTYUIOP ASDFGHJKL  ZXCVBNM'' '22*11/*.{32|}%+:Z 2$?0>2+R=~}*}%

The code can be tested here.

{                      # loop over all characters
  10{rand}:R~!         # take random number [0..9] and negate (i.e. 10% chance of true)
  {                    # {...}* is the if-block
    [                  # Three possible operations (code blocks) in the arry
      {.}              # a) duplicate
      {;}              # b) delete
      {                # c) shift
        Z              #      Z is the keyboard layout (see below)
        \?             #      Find the index of the current letter
        [.(.10-@).10+] #      Calculate index of letter left, right, above, below
        {Z=}%          #      Extract the corresponding letters for indices
        ' '-           #      Remove any spaces
        .,R=           #      Take random item
      }
    ]
                       # Z is the keyboard layout (upper and lower case)
                       # with enough spaces around
    'QWERTYUIOP ASDFGHJKL  ZXCVBNM'' '22*11/*.{32|}%+:Z
    2$?0>              # Is the current letter contained in Z and not a space?
    2+                 # Add 2 (i.e. 3 for letters, 2 for any other char)
    R=                 # Take a random code block from above
    ~                  # Execute the block
  }*
}%

PHP, 326 chars

$h=array(qs,vhn,vxf,sefx,wrd,drgc,fthv,gyjb,uko,hukn,jilm,ok,nk,bjm,ilp,o,aw,etf,awdz,rgy,yji,cgb,qse,zdc,thu,sx);$a=$argv[1];for(;$d=$a[++$b];)echo!rand(0,9)&&($c=ord($d)-65)?(($e=($c>-1&&$c<26))||($c>31&&$c<58)?(($g=rand(0,2))==0?$d.$d:($g==1?"":$h[$c-($e?0:32)][rand(0,strlen($h[$c-($e?0:32)]))])):(rand(0,1)?$d.$d:"")):$d;

And a more readable and commented version:

// $a   input
// $b   char iterator
// $c   current char ascii value
// $d   current char
// $e   is uppercase
// $g   rand() output
// $h   char displacement

// the neighbouring characters of a-z, in alphabetical (and ASCII) order
$h=array(qs,vhn,vxf,sefx,wrd,
    drgc,fthv,gyjb,uko,hukn,
    jilm,ok,nk,bjm,ilp,
    o,aw,etf,awdz,rgy,
    yji,cgb,qse,zdc,thu,
    sx);
$a=$argv[1]; // input from argument

for(;$d=$a[++$b];)
    echo!rand(0,9)&&($c=ord($d)-65)? /* 10% chance, read char ASCII value - 65 into $c */
        (($e=($c>-1&&$c<26))||($c>31&&$c<58)? /* is alphabetical? + is uppercase in $e */
            (($g=rand(0,2))==0? /* $g = 0 to 2, = 1/3 chance */
                $d.$d: /* double char */
                ($g==1?
                    "": /* omit char */
                    $h[$c-($e?0:32)][rand(0,strlen($h[$c-($e?0:32)]))])
                        /* a random character from the character's neighbours */
            ):
            (rand(0,1)?
                 $d.$d: /* double char */
                 "")): /* omit char */
        $d;

Still could be improved, I suppose.

JS, 303

function z(s){return s.split('').map(function(d){if((r=Math.random)()>.1)return d
if((c=r()*(/[a-z]/i.test(d)?3:2))<2)return c>1?d+d:''
if((x=(b="qwertyuiop0asdfghjkl000zxcvbnm").indexOf(d))<0)x=(b=b.toUpperCase()).indexOf(d)
for(a=0,n=11;i=!a;a=b[x+n])for(v=r()*4;v>i;n=i++%2?-n:1);return a}).join('')}

Algorithm for the keyslip:

• find char in string (or uppercase)
• add 11, -11, 1 or -1 to the index.
• if it's invalid (0 or null), re-roll