Can years of prednisone cause dental problems?

Yes: Yes there is potential though I have not seen full study on this. Prednisone do have side effect of osteoporosis which is thinning of bones. And so it can theoretically affect the bones of the facial structures and the mouth.

Prednisone Effects: Prednisone has several negative effects that can lead to dental problems. It does inhibit bone formation so any active periodontal disease can have accelerated bone loss. It raises blood sugar levels and that can worsen periodontal bone loss, gingivitis and increase tooth decay. It also causes changes in your blood lipid profile which can increase atherosclerosis leading to stroke & heart attack.

Possibly: Prednisone use affects your body's immune system negatively and if you have an active infection in the mouth (periodontitis, abscess, etc), it can affect your body's ability to fight off and contain that infection.