In terms of your original question title - you can't. The purpose of die and exit are to terminate script processing. If you don't want to terminate script processing, don't call die or exit.
In terms of suppressing the output of errors, this is possible using the error control operator (@) but in-advisable. (You must check for error state yourself if you're suppressing errors in this manner.)
For example, you could use:
$result = @mysql_query($sql);
if(is_resource($result)) {
// The query worked...
}
else {
// Handle error state, perhaps using mysql_error
}
However, just to make this clear, NEVER casually add the error suppressor to a function call. You MUST ensure you're handling any potential errors correctly yourself.