That depends on a few things. Are you still friends with your ex-girlfriend? If so, then absolutely keep her on your Facebook friend list and as a Twitter follower. I believe that women and men can be friends — like actual, platonic friends.
And if your new love interest has a problem with an ex posting friendly things on your Facebook page, then she’s most likely insecure about herself or insecure about your relationship. Nobody should ever tell you who to be friends with. Remember that.
On the other hand, if things didn’t end well with you and your ex-girlfriend, then it’s probably best to cut ties all together. You don’t want to be away from your Facebook page for more than five hours and come back to a 10-posting string of Tori Amos videos. Do you? Deleting exes off of your social media is not a black and white topic.
Your decision should be gauged on how things were left.