Probably. Yes, because the nose ring is made of metal and would should up on the x-ray as a white ring which may cover or block the tooth / root that your dentist may need to see.
Not if removed! Assuming the nose ring is made of metal or another material detected by a radiograph the short answer is yes. Especially if a film is being taken of a upper front tooth, the ring may obscure a dental issue or may create ghost like artifacts that may do the same. However, for pictures of just your lower teeth or bitewing "check-up" pictures of just your back teeth the ring wouldn't need removal.
Potentially. It can obscure view of an area, making it impossible to tell if there is a problem. Remove it for the x-ray and then replace it if you are so inclined.
Possible. It could interfere depending on what area is needed on the x-ray. Ask your dentist if you are unsure.
On some x-rays. Periapical or bitewings x-rays of the back teeth would not be affected or of the lower front teeth. But the upper front teeth due to angulation of the x-ray beam would be affected. Another x-ray call the panorex always require removal of hearings and any facial piercings. They cast bright spots on to the human tissues meant to be evaluated masking potential abnormalities. Cooperate, remove them.
Possible. The nose ring will appear on the xray in an area that can block a possible tooth image. However its up to the clinician to correctly interpret the xray. Its always better to remove if possible prior to taking the xray.