I've developed a close relationship with my past graduate teaching assistant, and I'm very interested in her: is it appropriate to pursue this? The graduate teaching assistant I'm interested in taught one of my introductory courses. She needed research assistants, so I decided to join her lab. From all this we became very close. Ever since the end of that semester, we talk nonstop.  It's been more than a year since I was in her class, but I am still an undergraduate. I am also applying to essentially the same graduate program she is in. We are not working with each other right now, but it is extremely likely we will work together again.
Would it be inappropriate to pursue this? The people in her lab know me. Would this make her look bad?
Since you are not working under her supervision right now, then assuming you are both adults, you are in the clear ethically, legally, and university policy-wise.
The only real caveats are that if you end up involved in a romantic relationship with her, she will not be able to write you a letter of recommendation, and will not be able to assume a supervisory role over you in the future. However, academia has plenty of dating and married couples working in the same department and sometimes on the same project. There are well-established norms for dealing with such situations, and no one thinks this is anything to get worked up about.
Good luck!
Edit: as I somewhat feared would happen, people are volunteering unsolicited advice about whether it’s a smart idea to enter a relationship with someone you may end up sharing a workplace with in some hypothetical future, based on further hypotheticals about what would happen if you end up splitting up or whatnot. Since I respect you as a fully autonomous adult capable of making your own decisions, I have restricted my answer to things you actually asked about and that are in the scope of the sort of advice this site is designed to offer. I would advise others to do the same. There’s nothing inappropriate about a romantic relationship in your situation. It may be a good idea or a bad one, wise or unwise based on numerous factors as with any question about whether two people are a good match for each other. But that’s not what you asked about, and has no connection to academia.