“Can a crown cause gum pain?”

Not usually. Some post op discomfort right after the placement of the crown is normal, but should go away after a few days

Yes, for many reasons. You will need to have it looked at, with an X-ray.

Hello, thanks for your question. Is very common to have gum sensitivity after getting a crown. The reason for it is the majority of the time in order to take a good model of your tooth to send to a lab to get your crown created we have to push your gums a little bit away from your tooth to have the most accurate impression. Also during the time you are having the temporary crown is very easy to have some inflammation of the gums. Make sure to floss and do some rinses with 1 tbsp of sea salt in 8oz of water twice a day, that should help until getting the permanent crown or even after if the case. For any persistent discomfort please call your Dentist for a follow-up. I hope I was able to help with your concern.

Dental crown can cause gum pain due to many reasons, such as excess cement, cementing crown on an inflamed gum, or violating the biologic width principal, among the host of many other reasons! It is best to call your dentist asap, for a check with xrays!
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If you just had the crown procedure done yesterday, your gums could be sore from part of the process that involves packing a cord into the gum crevice to accomplish a good impression. Use some warm salt water rinse 3x/day and take NSAIDS (Ibuprofen, naproxen, etc) to reduce the temporary inflammation and soreness. It should feel better in a few days or contact your dentist if it doesn't.