A door popping open in a GA aircraft is not considered an emergency. I have not seen it in the FAR/AIM. I don’t have a POH in front of me. If memory serves me correctly, the POH for a 2015 and later Piper Archer has this event listed under the category of Abnormal Procedures. There is a separate category for Emergency Procedures. On the checklists for three different aircraft models from my former flight school, it was also listed under Abnormal. Although the checklists are less than official documents confirming this event’s status, they definitely point to it not being an emergency.
Like the previous poster, I have had doors open during flight. It has always been a non-event. The doors tend to be designed with the hinges positioned on the leading edge of the door. This keeps the door from flying fully open in the aircraft’s slipstreams. One instructor actually demonstrated the effects of alternately opening the doors on a Cessna 172 to induce yaw. You have to apply a considerable bit of force to swing the door fully open at 90 knots.
Conversely, only half of the Robinson and Loach helicopters I have flown have had the removable doors in place. And, most of the smaller skydiving planes in which I have flown have been unmodified, regular GA planes with the seats removed. Some of them had the right door removed, also, and were still able to achieve 17,000 feet. Opening the door (or in some cases, the tail) does not seem to adversely affect performance enough to classify it as an emergency.