Would removing spaces in a string protect against SQL injection?

No. Removing spaces would not prevent SQL injection, as there are many other ways to make the parser process your input. Lets look at an example. Imagine that you had a url which used user supplied input unsafely in a query:

http://example/index.php?id=1 => SELECT * from page where id = 1

In your example the attacker would use spaces:

http://example/index.php?id=1%20or%201=1 => SELECT * from page where id = 1 or 1=1.

Removing the spaces would collapse the injection into a string.

Now let's imagine that the attacker used another form of white space, i.e. tabs:

http://example/index.php?id=1%09or%091=1 => SELECT * from page where id = 1 or 1=1.

Removing the spaces would still allow the injection through.

Depending on the technology in use the attacker could replace the spaces with /**/ %00 %09 %0a %0d or any number of unicodes that causes tokenization by the SQL parser. While the referenced example removed more than just spaces, the aforementioned example takes advantage of SQL comments to cause the tokenization which are not whitespace. You would still be vulnerable.

The only reliable way to prevent SQL injection is to use parameterized queries.

We're in 2016! SQL injections are a thing of the past unless you use insecure code.

Whatever language you use, if you want to prevent any and all SQL injections, use prepared statements or any other type of data binding.

Prepared statements separate the query from the data, making it impossible for the data to affect the query.

To directly answer your question, removing spaces would reduce SQL injections (when using outdated code and libraries), but would surely limit your input text (no spaces anywhere).

It would limit the problem, but not eliminate it. Consider the following situation:

$un = str_replace(" ", "", $_POST["username"]);
$pw = hash($_POST["password"];
$sql = "SELECT * FROM users WHERE username = '$un' AND password = '$pw'";

Lets say I post the username admin'--. That would log me in as admin, without using a single space.

As wireghoul points out, you would need to remove other blank characters like tab as well. But as Julie Pelletier points out, just use prepared statements. Trying to come up with clever schemes to stop SQLi without it might be a fun game, but it will never give you the security that prepared statements does. Everything else is just a distraction.

No. Let's say you have this as your SQL:

"select * from people where last_name = '" + surname + "'"

If I enter: 'OR(1=1)OR'a'=' into the input it turns into:

select * from people where last_name = ''OR(1=1)OR'a'=''

Which executes in Oracle and returns all the rows from the table.