Will my fillings hurt after novocain (procaine) wears off?

Short answer, "No" You might have some residual discomfort after the "novocaine" wears off. It might be due to the actual injection and/or the drilling of the tooth, but if there is any discomfort, it should resolve very rapidly. If not, go back to your dentist. Sometimes the bite needs a little refinement, or there are other underlying problems that have to be addressed, especially in deep, extensive fillings.
You will feel sore. You may feel a little bit of soreness and sensitivity to hot and cold temperatures immediately after the novocain (procaine) wears off and maybe for a day or two. Any discomfort is very temporary and will quickly wear off. Take Ibuprofen or any otc pain medication in the meantime.

Related Questions

Does the novocain (procaine) shot hurt for cavity filling?

No. Your dentist will most likely use a topical anesthetic to numb the gingival tissues prior to injection.
Thanks for asking! You would fill a pinch with the shot as well as some pressure/burning in the immediate period when the local anesthetic is injected. After that, it will be much better and you won't feel much of anything.