Will my lips take longer to stop swelling and risk infection if I continue taking my antibiotics? Do I stop taking them for now?

Having an active infection during injections is not ideal, but if you are being treated for it, the risk is much less. I'm not aware of any way that the antibiotics you are taking would negatively affect the injection. I would continue taking the antibiotics as prescribed. If you experience pain and swelling in the lips that is worsening or looks like an infection, then certainly reach out to your doctor to see if more antibiotics are needed.

I don't think it is a good idea to have an elective procedure while an active infection is going on. I would've been best to complete the course of antibiotics prior to treatment. Now that you have had injections, it is even more important to finish them, don't stop them. There is a risk of infection of the filler because the bacteria can spread to the filler.

Hi and welcome to our forum!
Definitely be on the alert for increasing swelling, pain, redness, drainage, or fever. Although the risk for infection is small, that risk is increased 7x if there is a concurrent infection elsewhere. As you are already under treatment, that risk is somewhat lessened. Do not stop your antibiotic. So far, you have played Russian roulette with empty chambers.
Best wishes...