You can do it like this:
string tmp = "Hello 'World'";
tmp.replace("'", "");
But that will just replace single quotes. To replace double quotes, you must first escape them, like so:
string tmp = "Hello, \"World\"";
tmp.replace("\"", "");
You can replace it with a space, or just leave it empty (I believe you wanted it to be left blank, but your question title implies otherwise.