I would let the dentist know either way.
I am apparently sensitive to Novocaine, because it takes that stuff FOREVER to wear off for me (although not until the next day), and also I am terribly sore from the shot for several days. I find I can't eat anything for 3-4 days after Novocaine. So, I've started getting only Nitrous Oxide for fillings, and wow what a difference. As soon as the procedure is over, I have no pain at all! I'm DONE! The small amount of pain I feel durng the procedure is very preferable to being unable to even drink water for 6-8 hours afterwards and then being on basically a liquid diet for 3 days.
If I do have something done where it's not really possible to go without Novocaine, I ask them to use a smaller dose, and that helps too.