In general when you want to stop a module or function from dying, you wrap the call in an eval{ } block. Then you can test $@ to see if an error occurred. For example:

eval {
    $msg->send();
};

if($@) {
    # We had an error, log it to a file...
    $log->add_message($email_address, $@);
}
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This prevents the script from dying if send(); fails. Of course, you want to make sure sending is working properly to begin with (maybe your script is failing for some reason other than a bad email address - you can print the value of $@ to see why it is failing).

Update: Uh, let me clarify. That will catch the $msg->send() function ONLY IF it calls die() (raises an exception). So if the send() function returns true or false based on success or failure, you should check other solutions in this thread.