The Stratospheric Observatory for Infrared Astronomy (SOFIA) is a modified 747 that has an 5.5 m x 4.1 m door that is opened during flight for the installed in infrared telescope.
But the door is usually closed during takeoff and landing.
But according to this story there are emergency procedures to land with an open door and it had to do so once, when it became stuck half open.
Images courtesy Wikipedia
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Aloha Airlines Flight 243 proved that it's possible for a plane to fly with about 25% of its roof missing.
The door, however, appears to have remained closed until landing.
Image courtesy Wikipedia
aviation.stackexchange.com/users/5517/FreeMan
Check out my answer here to a similar question (about what would happen if you open a door in flight).
In short it depends largely on the plane but in the most general way of looking at it.
You could do it up until you would need to pressurize the plane (or you would have wear O2 masks or something to be able to breathe).
UA Flight 811 was able to make a safe decent and return to airport after its door blew out so large planes are maneuverable with out door.
Planes like the DC-3 used in WWII for paratrooping regularly ran with their doors open.
If you consider rear cargo ramps doors, the C-130 can open its rear ramp for HALO jumping at high altitudes .
In typical HALO/HAHO insertions the troops jump from altitudes between 15,000 feet (4,600 m) and 35,000 feet (11,000 m)
There is at least one other powered glider I can think of that can open its doors mid flight...
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Since comments mentioned that not all of the outcomes from Aloha 243 were exactly safe, here's another example that actually was quite safe: Southwest 812 had a large hole open in the roof during flight at cruise altitude, resulting in rapid depressurization at 34,000 ft. 2 of the 123 people on board (a flight attendant and a passenger)
suffered minor injuries.
The plane diverted to Yuma, Arizona and landed safely about 26 minutes after the depressurization.
As with the other cases, the doors were technically shut, but I'm not so sure that that made much difference.
Furthermore, this was not the first time this had happened.
Less than 2 years before this incident, Southwest 2294 had a similar incident (though with a smaller hole) and safely diverted to Charleston, WV with
no injuries at all.
What's more:
according to the FAA's registry and airfleets.net , it looks like both of these aircraft were returned to service and are still actively flying for Southwest!
So, it looks like both the "you can walk away from it" requirement and the "the aircraft is reusable" preference were met in both instances.
The Boeing 727 Another case that actually doesn't involve any structural damage at all is the 727 and its airstair.
In the famous case of D.B. Cooper's hijacking , this occurred while the aircraft was pressurized.
According to the wiki on the incident, the result was the following: The crew soon noticed a subjective change of air pressure, indicating that the aft door was open.
At approximately 8:13 pm the aircraft's tail section sustained a sudden upward movement, significant enough to require trimming to bring the plane back to level flight.[33]
At approximately 10:15 pm Scott and Rataczak landed the 727, with the aft airstair still deployed, at Reno Airport.
Additionally, some skydiving clubs even offered dives from the 727 for many years, though these presumably didn't pressurize the cabin.
I'm not sure whether this is still offered.
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Yes they can, here's a Convair 880 with cargo door open.
DC-8 with cargo door open.
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For the whole flight envelope, maybe not.
Depends on the door design.
Gear doors frequently have speed limits before they fly off on their own.
Large openings create substantial drag, so we may find that the envelope is self-limiting: the aircraft can no longer reach speeds and/or altitudes where the opening is a big problem.
There are too many variables for a simple answer.
aviation.stackexchange.com/users/1632/paul
Biplanes and ultra-lights don't have any doors at all.
In general, for any plane the wing, the empennage and, to a lesser extent, the surface area on the lower part of the fuselage is all that is important.
If you installed structure-reinforcing crossbars you could remove the entire fuselage wall of a commercial jet and it would still fly.
aviation.stackexchange.com/users/1378/Tyler Durden
From another  question  asks about the possibility of dropping bombs from converted airliners. My question: is it possible to safely fly aircraft with a door open for the whole flight envelope? Would the aerodynamics be affected if we did not close the door?  I am not asking about the need of oxygen for the crew, just about the aerodynamics involved. I know the plane would not be pressurized.
Is it possible for an airliner to safely fly with doors open?
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<aerodynamics><airliner>
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Planes can fly with open doors as cases have been documented and some planes are meant to have an opening, for example planes used for paratroopers. It should be noted that large openings create substantial drag therefore speed/altitudes where big openings are a problem cannot be reached. A yes or no answer to this question is not possible because there are too many variables. One consideration is that a large opening would produce a lot of drag. As a result, the aircraft would not reach a speed or altitude where the opening would cause major problems. Examples are cited of aircraft flying with doors open, either intentionally or as the result of an accident. 
There are known examples of aircraft flying with doors open, either intentionally or as the result of an accident.  A large opening would produce a lot of drag. As a result, the aircraft would not reach a speed or altitude where the opening would cause major problems.  A yes or no answer is not possible because there are too many variables.