Would it be wrong to tell my friend to stop venting to me about her breakup?
Are you worried that will cause a rift with her?
I worry that she'll think I'm not a caring person.
It can be tedious dealing with a friend who seems to be going through the same thing all the time, but we need to be supportive of our friends especially when they're going through a break up. If your friend is constantly talking about her break up, it's because it's something that's stressful for her and she trusts you to listen. If you tell her to stop talking about the thing that stressing her out, then she won't trust you for support in the future.
What if she says something mean to me during her venting? Should I just listen to that too?
No, that would be wrong on her part. She shouldn't be mean to you just because she's stressed about her breakup. You wouldn't be wrong to call her out on it if she is.