Can planned runway closures prevent a plane from leaving an airport?
There is actually a really good example of this...

In 1994 it was decided that the infamous Meigs Field would be demolished and turned into a park. (I say infamous because for all of us young flight simulator aficionados this is the home airport of Microsoft Flight Simulator). 

On a Sunday night, March 30, 2003, the Mayor Daley of Chicago ordered the runway bulldozed by gouging huge X's into the runway itself. This was done in the middle of the night with many aircraft still based on the field. No notice was given to the FAA or the owners of aircraft still on the field. In fact, an inbound flight had to be diverted by air traffic control because of equipment on the runway.

You can imagine what kind of excrement storm this stirred up. In the morning of March 31, 2003 at a press conference Daley defended his actions by saying that it would have saved the city of Chicago years of court battles over closing the airport. Luckily the aircraft at the field were allowed to depart from the 3000' taxiway that ran adjacent to the runway.

In this case, the worst the FAA could do was fine the city of Chicago \$33,000 (\$1,100 per day) for closing an ILS procedure without a 30 day notice. Later it was found that they misappropriated FAA funds to demolish the runway and build the park and that had to be paid back, but there were no legal implications of this. The airport was not under federal funding and the city was technically within its rights to close it.

You can read more about it on Wikipedia

The point is that airports are often owned by the city in which they reside, or sometimes the state, at least in the U.S. The FAA releases many guidelines for airports to make them safe, and even has an "airport certification program" for certifying part 139 operations, however these are done through "orders" and guidelines, not much in the way of FAR's. 

ATC, if federally operated, must comply with the regulations that apply, including giving NOTAM information for planned closures, or even NOTAM's for closures that are not planned, such as runway debris or accidents closing runways. 

So really when you get down to it, the city that owns the airport can close runways (or the entire airport) whenever they want, and yes this may strand aircraft. Usually if the airport is closed unexpectedly and aircraft are still there, they will be allowed to take off from taxiways or grass strips. Worst case scenario the wings will have to be removed and it will need to be trucked out, at least for GA operations. 

Depending on the Ops Spec for the airline, it may also allow the aircraft to take off from a taxiway. Usually something like that is done without passengers on board though. 

So to address your specific questions...

Is an airport required to give notice to pilots (or aircraft owners) about planned construction? No, there is no requirement for this. Usually what happens is airports don't give adequate notification and it agitates owners, but it all works out in the end.

For example, can an airplane ever be stranded while airport construction is occurring? Yes, at Meigs originally owners had to fight to be allowed to take-off. Many aircraft where going to be scrapped as they were too expensive to move by truck, and owners didn't want to pay. The city didn't want to pay, and the insurance company didn't want to pay, so that would have kicked off a big set of lawsuits for the city.

Closing runways can't always be avoided, for example when there is an accident, weather (flooding, snow, ice), earthquakes, etc. It happens and the system "works" around it when it does.