My urinary tract infection seems to be gone. Do I still need to take the rest of my antibiotic medication?

Yes. Excellent that you are no longer symptomatic, but does not mean that UTI has completely resolved. Antibiotics will also give you some protection from getting reinfected whilst bladder heals. You are at increased risk of getting another UTI during the healing phase. So it's a good idea to complete the course of antibiotics.
Certainly. Subsidence of symptoms, e.g., painful urination or fever, does not generally mean complete clearing of uti. Premature termination of drug treatment results in leaving half-dead (antibiotic resistant) bugs behind.

Related Questions

How soon after finishing my anitibiotics for a urinary tract infection can I have sex?

I believe anytime. Unless there is a specific issue with your condition, the type of antibiotic you are using or some very unusual problem, taking antibiotics should not preclude one from having intercourse. If a vaginal or penile condom is used, I cannot think of a reason otherwise. Take the antibiotics as prescribed and take care of yourself. See your doctor otherwise (call or return to clinic).
As soon as you like. As soon as symptoms of pain and discomfort from UTI have settled down. Not if bladder pain is still present or if you experience pain with sex.

How soon after I finish my anitibiotics for a urinary tract infection can I have sex again?

When you feel like. It. Just be on the lookout for a recurrence so that you may treat it promptly if it happens.