Are planes allowed to fly through a planned professional fireworks display?
Most fireworks go no higher than 1200 ft and consumer sized stuff is more around 500ft and below. which is fairly low for most regimes of flight for all but general aviation planes and planes taking off/landing. With this in mind your main point of concern would be shooting them off near an airport, or people flying low and slow near your display. 

I cant find any regulation explicitly forbidding it although its unwise and it most likely would be considered "careless or reckless operation as per §91.31" if you flew directly (and deliberately) into a display at low altitude, Fireworks are noted in the NOTAM's and it is the pilots responsibility to make themselves aware of all flight risks and en-route information as per § 91.103 Preflight action. You can see a good example here in todays NOTAMs listing various firework displays and their locations. 



(source)

Its also the responsibility of the launcher to file the NOTAM in the first place.