Is it normal to have throbbing and heat sensation after a Juvederm lip injection? (photos)

How long ago did you have them injected?  This could be normal swelling after injection , the lips are very sensitive.   It's possible it could be a reaction.  Call the office where you had them injected and speak to the provider.

CALL the injecting doctor ASAP. Sounds like an allergic reaction or a break in technique with possible infection/cellulitis. But are you "sure" it was USA Juvederm??? No, Urgent Care DOES NOT carry hyaluronidase enzyme to dissolve HA fillers. But if they did they would need verification you had an HA filler from the injecting doctor.. Also the costs for the hyaluronidase is expensive like $800 fort injection.  

From your description you could have a blocked artery from the Juvederm. This can cause excessive pain, but I don't see any of the other telltale signs in your photo..... No, a trauma unit or ER or QuickCare would not have the enzyme to dissolve Juvederm. You need to find a cosmetic physician of some kind that can see you and dissolve the filler if needed. "This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."