For me, with a potential friendship, I'll walk away if the person has different values/beliefs in certain hot button areas, like politics, religion, etc. I have befriended several people who didn't share my beliefs, and at some point, a conflict always arises. I actually had a good friend dump me because I didn't share her religious beliefs, and I can't tell you how much that hurt.
As for people who I have already befriended, neediness is a biggie for me. I had to end contact with a friend a few years ago because her neediness was interfering with my life. She had a lot of emotional problems, and she would call almost daily, crying and complaining, and just would not stop. I would try to be polite, and spend a half hour telling her that I had to get off the phone, and she wouldn't hang up. I had to start getting tough with her, and would just say that I was late for an appointment, and would hang up on her. Then she would call back and leave a long message on my machine telling me how much I had hurt her. I finally stopped taking her calls and emails, and had to end the friendship. I have enough problems of my own - I can't be someone's therapist. If the person is willing to reciprocate, and listen to my problems as well, that's different. But when it's completely one-sided, and I'm being expected to provide all of the support, I just can't do it.