Will guys date a girl their friends don't approve of?

Depends on why they don't approve of her. If they have a history with her and know her personally then I'll want to know why. Example:

Back when I was in high school I had a crush on a girl named Courtney who was pretty, very shy and reserved, and seemed like an overall cool person even though I didn't personally know her that well. It turned out she was an ex girlfriend of my friend Eric and also of my friend Brandon. They both ended up breaking it off with her because they said she was extremely clingy, possessive, and controlling. That was enough for me to move on.

But if there is no history like that and they just "don't like her", f*** 'em. I'll date who I want and unless they can tell me why she's trouble and offer up proof, I'll just let it go in one ear and out the other.

I would. ... I don't ask my friends for permission to date anyone. If I like a girl, that's good enough for me. ;-)

Well, to get close to a guy you have to get close to his buds. Then the guy's friends will probably get to know you more and get to like you. When a guys friends approve then its basically a guaranteed spot in girlfriend.