JavaScript (ES6), 34 bytes

f=(d,a)=>a.reduce((r,i,j)=>r*d[j]+i)

Surely reduce must be better than map.

Python, 43 bytes

f=lambda x,y:x>[]and y.pop()+x.pop()*f(x,y)

Test it on Ideone.

Jelly, 7 6 bytes

Ṇ;żḅ@/

Try it online! or verify all test cases.

How it works

Ṇ;żḅ@/  Main link. Arguments: D (list of dimensions), I (list of indices)

Ṇ       Yield 0, the logical NOT of D.
  ż     Zip D with I.
        If D = [10, 10, 4, 62, 7] and I = [1, 2, 3, 4, 5], this yields
        [[10, 1], [10, 2], [4, 3], [62, 4], [7, 5]].
 ;      Concatenate, yielding [0, [10, 1], [10, 2], [4, 3], [62, 4], [7, 5]].
   ḅ@/  Reduce by swapped base conversion to integer.
        [10, 1] in base    0 is    0 × 10 + 1 = 1.
        [10, 2] in base    1 is    1 × 10 + 2 = 12.
        [ 4, 3] in base   12 is   12 ×  4 + 3 = 51.
        [62, 4] in base   51 is   51 × 62 + 4 = 3166.
        [ 7, 5] in base 3166 is 3166 ×  7 + 5 = 22167.

Pyth, 10 bytes

e.b=+*ZNYC

Try it online: Demonstration or Test Suite

Using Horner's method to calculate the index.

J, 2 bytes

#.

Where there's an APL, there's a J! Kind of. Takes dimensions as left arg and index as right arg. "Indexing a multidimensional array is essentially mixed base conversion."

MATL, 11 bytes

4L)1hPYpP*s

This uses 0-based indexing, as in the original challenge.

Try it online!

Explanation

The code explicitly does the required multiplications and additions.

4L)    % Take first input array implicitly. Remove its first entry
1h     % Append a 1
PYpP   % Cumulative product from right to left
*      % Take second input array implicitly. Multiply the two arrays element-wise
s      % Sum of resulting array. Implicitly display

MATL, 9 bytes

PiPZ}N$X]

This uses 1-based indexing (now allowed by the challenge), which is the natural choice in MATL.

To compare with the test cases in the challenge, add 1 to each entry in the input index vector, and subtract 1 from the output.

Try it online!

Explanation

The code is based on the builtin X] function, which converts multidimensional indices to a single, linear index (like Matlab or Octave's sub2ind function).

P      % Take dimension vector implicitly. Reverse
iP     % Take vector of indices. Reverse
Z}     % Split vector into its elements
N$X]   % Convert indices to linear index (`sub2ind` function). Implicitly display

Julia, 29 27 bytes

x%y=prod(x)÷cumprod(x)⋅y

Try it online!

Python, 85 bytes

lambda a,b:sum(b[i]*eval('*'.join(str(n)for n in a[i+1:])or'1')for i in range(len(a)))

I'll probably get my butt kicked by the better python golfers out there.

Python 3.5, 69

lambda d,i:sum(eval("*".join(map(str,[z,*d])))for z in i if d.pop(0))

Test here

Haskell, 34 bytes

a#b=sum$zipWith(*)(0:b)$scanr(*)1a

Usage example: [10,10,4,62,7] # [1,2,3,4,5] -> 22167.

How it works:

      scanr(*)1a  -- build partial products of the first parameter from the right,
                  -- starting with 1, e.g. [173600,17360,1736,434,7,1]
    (0:b)         -- prepend 0 to second parameter, e.g. [0,1,2,3,4,5]
  zipWith(*)      -- multiply both lists elementwise, e.g. [0,17360,3472,1302,28,5]
sum               -- calculate sum

Octave, 58 54 bytes

Thanks to @AlexA. for his suggestion, which removed 4 bytes

@(d,i)reshape(1:prod(d),flip(d))(num2cell(flip(i)){:})

Input and output are 1-based. To compare with the test cases, add 1 ot each entry in the input and subtract 1 from the output.

This is an anonymous function. To call it, assign it to a variable.

Try it here.

Explanation

This works by actually building the multidimensional array (reshape(...)), filled with values 1, 2, ... in linear order (1:prod(d)), and then indexing with the multidimensional index to get the corrresponding value.

The indexing is done by converting the input multidimensional index i into a cell array (num2cell(...)) and then to a comma-separated list ({:}).

The two flip operations are needed to adapt the order of dimensions from C to Octave.

Haskell, 47 bytes

Two equal length solutions:

s(a:b)(x:y)=a*product y:s b y
s _ _=[]
(sum.).s

Called like: ((sum.).s)[4,2][5,10].

Here's an infix version:

(a:b)&(x:y)=a*product y:b&y
_ & _=[]
(sum.).(&)

Mathematica, 27 bytes

#~FromDigits~MixedRadix@#2&

An unnamed function which takes the list of indices as the first argument and the list of dimensions second. Based on the same observation as Dennis's APL answer that computing the index is really just a mixed-base conversion.

CJam, 7 bytes

0q~z+:b

Try it online!

How it works

0        e# Push 0 on the stack.
 q       e# Read and push all input, e.g., "[[10 10 4 62 7] [1 2 3 4 5]]".
  ~      e# Eval, pushing [[10 10 4 62 7] [1 2 3 4 5]].
   z     e# Zip, pushing [[10 1] [10 2] [4 3] [62 4] [7 5]].
    +    e# Concatenate, pushing [0 [10 1] [10 2] [4 3] [62 4] [7 5]]
     :b  e# Reduce by base conversion.
         e# [10 1] in base    0 is    0 * 10 + 1 = 1.
         e# [10 2] in base    1 is    1 * 10 + 2 = 12.
         e# [ 4 3] in base   12 is   12 *  4 + 3 = 51.
         e# [62 4] in base   51 is   51 * 62 + 4 = 3166.
         e# [ 7 5] in base 3166 is 3166 *  7 + 5 = 22167.

C++, 66 bytes

A quick macro:

#include<stdio.h>
#define F(d,i) int x d;printf("%d",&x i-(int*)x)

Use like:

int main(){
    F([5][1][10], [3][0][7]);
}

This may be a bit of an abuse of the rules. Creates an array with the given size, than checks to see how far the given indexes offset the pointer. Outputs to STDOUT.

This feels so dirty... But I just love the fact that this is valid.

Mathematica, 47 bytes

Fold[Last@#2#+First@#2&,First@#,Rest/@{##}]&

(Unicode is U+F3C7, or \[Transpose].) For this, I rewrote the expression as D_n(D_n-1( ⋯ (D₃(D₂S₁ + S₂) + S₃) ⋯ ) + S_n-1) + S_n. Just Folds the function over both lists.

Actually, 13 bytes

;pX╗lr`╜tπ`M*

Try it online!

This program takes the list of indices as the first input and the list of dimensions as the second input.

Explanation:

;pX╗lr`╜tπ`M*
;pX╗            push dims[1:] to reg0
    lr`   `M    map: for n in range(len(dims)):
       ╜tπ        push product of last n values in reg0
            *   dot product of indices and map result

Octave, 47/43/31 bytes

@(d,i)sub2ind(flip(d),num2cell(flip(i+1)){:})-1

Test it here.

Having said that, as it was asked in a comment, 1-based indexing was said to be OK when this is natural to the language being used. In this case, we can save 4 bytes:

@(d,i)sub2ind(flip(d),num2cell(flip(i)){:})

In analogy, I argue that if the objective of the code is to linearly index an array within that language, the whole flipping around and accounting for MATLAB/Octave's column major order should not be necessary, either. In that case, my solution becomes

@(d,i)sub2ind(d,num2cell(i){:})

Test that one here.