Can novocaine at dentist cause swollen gums?

Gum soreness: It depends where the novocaine was put. If it is at or near the injection site then yes. Sometimes a clamp or spacer band it placed around/under the gum that can also cause soreness. If it is not gone in a day or two call your dentist.

Yes and no: A small amount of swelling at or near the injection site is not uncommon, and should quickly resolve. Sometimes the dental proceeders themselves can irritate the gums causing temporary swelling or sensitivity. If swelling does not resolve in a day or two, call your dentist. Know that localized swelling may indicate infected tooth, generalized swelling may indicate infected gums.