If I have a pc that is connected to the Internet via a NAT router but only use it for gaming and never to browse the web, is it safe to not bother with anti virus software or will just being connected mean I'm vulnerable to some kind of virus?
There's no need to use A/V if you know how to use a computer in a safe manner.
If in doubt, use A/V.
What do you class as using a computer in a safe manner? Even normally safe websites can deliver malware onto your PC, heck just recently Anandtech had a malicious ad banner and was putting Trojans on peoples PC's.