I've never had a license and never had a desire to get one. May be a deal breaker for some guys, but I've never had issues, luckily. I think once that they realize that it's not out of laziness that I don't drive, they're cool with it. Honestly, I'm scared to learn and never really been interested in owning a car. I remember being almost 16 and all my friends were really excited about taking drivers' ed and getting learner's permits and licenses and I never had that excitement.
I've never begged people for rides and have always enjoyed walking or taking the bus wherever I need to go. Although, it is a little tougher here. I lived in Chicago for four years and public transport is very convenient, just like New York City. Lots of people don't own cars there because parking is a hassle, it's expensive, and it's very easy to take the train or bus because they run very often. Now I live in a small town in Wisconsin and our buses only run once every half hour or hour. It still doesn't bother me though-just requires a bit of planning if I take the bus somewhere. And I love walking, except in the brutal winter.
I think it's definitely tougher for guys who don't drive than women though. I've never met a guy who was annoyed by my lack of license and car, although I'm sure there's some out there. I think a lot of women would be hesitant to date a guy who doesn't drive. Like others have previously mentioned though, a lot depends on where you live.