Example:

As you can see here, the regex consists of 2 alternating capturing groups. The first one captures all the quoted strings. The second one all the comments.

All we need to do, is removing everything captured by the 2nd group.

Example:

Python2 - 163 bytes

def f(s):
 import re
 for x in re.findall(r'("[^\n]*"(?!\\))|((?:\/\/[^\n]*$)|\/(?!\\)\*[\s\S]*?\*(?!\\)/)', s, 8):
  if x[1]:
   s = s.replace(x[1], '')
 return s

>>> def f(s):
...  import re
...  for x in re.findall(r'("[^\n]*"(?!\\))|((?:\/\/[^\n]*$)|\/(?!\\)\*[\s\S]*?\*(?!\\)/)', s, 8):
...   if x[1]:
...    s = s.replace(x[1], '')
...  return s
... 
>>> code = r'''#include <stdio.h>
... 
... int main(int argc, char** argv)
... {
...     // this comment will be removed
...     if (argc > 1) {
...         printf("Too many arguments.\n");   // this too will be removed
...         return 1;
...     }
...     printf("Please vist http://this.will.not.be.removed.com\n");
...     printf("/* This will stay */\n");
...     printf("\"/* This will stay too */\"\n");
...     printf("//and so will this\\");
...     // but not this
...     printf("just \"ano//ther\" test.");
...     return 0;
... }
... /*
...     this shall disappear
... */
... #include <string>
... int main(int argc, char** argv)
... {
...     string foo = ""/*remove that!**/;
...     // Remove /* this
...     int butNotThis = 42;
...     // But do */ remove this
...     int bar = 4 /*remove this*/* 3; // but don't remove that 3. */
...     return 0;//just a comment
... }/*end of the file has been reached.*/'''
>>> print f(code)
#include <stdio.h>

int main(int argc, char** argv)
{
    
    if (argc > 1) {
        printf("Too many arguments.\n");   
        return 1;
    }
    printf("Please vist http://this.will.not.be.removed.com\n");
    printf("/* This will stay */\n");
    printf("\"/* This will stay too */\"\n");
    printf("//and so will this\\");
    
    printf("just \"ano//ther\" test.");
    return 0;
}

#include <string>
int main(int argc, char** argv)
{
    string foo = "";
    
    int butNotThis = 42;
    
    int bar = 4 * 3; 
    return 0;
}

Python2 - 163 134 bytes

import re
def f(s):
 for x in re.findall(r'("[^\n]*"(?!\\))|(//[^\n]*$|/(?!\\)\*[\s\S]*?\*(?!\\)/)',s,8):s=s.replace(x[1],'')
 print s

Python 2.7.9 (default, Dec 11 2014, 04:42:00) 
[GCC 4.9.2] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> import re
>>> def f(s):
...  for x in re.findall(r'("[^\n]*"(?!\\))|(//[^\n]*$|/(?!\\)\*[\s\S]*?\*(?!\\)/)',s,8):s=s.replace(x[1],'')
...  print s
... 
>>> code = r'''#include <stdio.h>
... 
... int main(int argc, char** argv)
... {
...     // this comment will be removed
...     if (argc > 1) {
...         printf("Too many arguments.\n");   // this too will be removed
...         return 1;
...     }
...     printf("Please vist http://this.will.not.be.removed.com\n");
...     printf("/* This will stay */\n");
...     printf("\"/* This will stay too */\"\n");
...     printf("//and so will this\\");
...     // but not this
...     printf("just \"ano//ther\" test.");
...     return 0;
... }
... /*
...     this shall disappear
... */
... #include <string>
... int main(int argc, char** argv)
... {
...     string foo = ""/*remove that!**/;
...     // Remove /* this
...     int butNotThis = 42;
...     // But do */ remove this
...     int bar = 4 /*remove this*/* 3; // but don't remove that 3. */
...     return 0;//just a comment
... }/*end of the file has been reached.*/'''
>>> f(code)
#include <stdio.h>

int main(int argc, char** argv)
{
    
    if (argc > 1) {
        printf("Too many arguments.\n");   
        return 1;
    }
    printf("Please vist http://this.will.not.be.removed.com\n");
    printf("/* This will stay */\n");
    printf("\"/* This will stay too */\"\n");
    printf("//and so will this\\");
    
    printf("just \"ano//ther\" test.");
    return 0;
}

#include <string>
int main(int argc, char** argv)
{
    string foo = "";
    
    int butNotThis = 42;
    
    int bar = 4 * 3; 
    return 0;
}