Does hyaluronidase cause any swelling or redness at the injection site?

Possibly: Some people get allergic to hyaluronidase. It is used to dissolve fillers- hyaluronic acids. Give your filler a couple days to settle in before dissolving if you can as it is expensive to have fillers placed.

Possibly: Any injection can cause swelling and redness, as there is a puncture to the skin and can cause inflammation or infection.