Can the facial vein press on the infraorbital nerve causing tingling?

Comprssion/infaorbit. Possible but doubtful. Best to be seen in person by a Facial PS.
Facial veins. There are many branches of the facial vein but none of these should cause compression of adjacent nerves leading to numbness or tingling. Most likely the cause of your symptoms is not vein related. See a facial surgeon for an evaluation.