Whether a double's action could be damaged by a double discharge would, IMO, depend on several variables like...

How well the gun was made, ie how strong is the action design, how well are the barrels fitted to the frame, how good is the steel used and how well was it heat treated,how much wear has occurred in the bolting system over years of use, and how high is the pressure of the cartridge it is chambered for.

That said, any good quality double in sound condition and chambered for a low pressure cartridge like 450 No2 or 470 or similar should not have any problem with the action anyway.
Maybe the stock might have problems if it was doubling on a regular basis.

But say you had a crappy designed and made, badly worn double chambered in 375 H&H BELTED Magnum and shooting handloads that were running very high pressures, well, anything could happen I guess.

I know of a OU double in 500 NE built on a 12 guage frame that doubled a few times when it was first shot and it did not suffer any ill effects.
Some trigger work cured it of this nasty habbit.