Focusing on the exploits of the fictional Korean War US Army Mobile Surgical Army Hospital 4077th, the show has some common ground with MST3K : People making the best of a tense and often insane situation, and cast changes aplenty as the seasons went on, as well as spirited arguments about when the show was at its best and with which cast combination, as well as a stable of memorable supporting and recurring characters, including ones 'promoted' from recurring to regular. Cracking wise while avoiding cracking up was another large common area. Another is higher-ups making capricious and often ridiculous decisions often from out of nowhere.
The references to M*A*S*H were plentiful (and may be again with the late 2010's revival) but one joke could be counted on : Any scene in a movie involving a helicopter would trigger some manner of reference to the show's opening credits, involving the flight of choppers into the medical compound.
List of M*A*S*H references on MST3K
(TBC... This one may take some time to compile)