Can a problem occur if a dental crown is not fully seated against the gumline?

Usually what you are feeling is
the junction of the crown where it meets the natural tooth. Usually a
dentist, unless there was a cavity here, will leave the edge of the new crown
slightly above the gum line on the inside. There is always a possibility
that the dentist did not put the crown all the way down and there is a
gap. I would return to your dentist and have it checked.

Yes, there can be catastrophic results like caries and decay of the underlying or adjacent tooth, Gum problems, Bone loss, Etc. You should get it checked with an X-ray as soon as possible and if the marginal fit is questionable, get a new crown done.