Base Breaker: Both Willie and Short Round get a large amount of this for different reasons.
Willie gets hate for being a very unhelpful damsel with a tendency to scream, although some do not mind, pointing out that either she wasn't as annoying in hindsight or that she served as an interesting alternative to Marion's Action Girl and Elsa's traitorous streak.
Big Lipped Alligator Moment: The opening Busby Berkeley-esque dance number, which is weird in itself, but certain elements of the number ramp this up even further — for instance, the huge sound-stage the dancers disappear into, which is apparently somewhere inside the club, the fact that they can all apparently rise back into a standing position from a grounded scissor-split, and when Willie somehow turns several red hankerchiefs plucked from the dancers' lapels into a huge red sheet (well, she does mention her grandfather was a magician....). All in all, very strange indeed.
Complete Monster: In a series full of villains that include Nazis, the evil Thuggee high priest Mola Ram stands out. Head of the cult of Kali Ma, Ram intends to find the Sankara Stones to bring forth the reign of Kali. To find the stones, he has children abducted from their villages and enslaved. Ram has a habit of ritualistic killings by ripping the hearts out from his victims and dipping them in lava. Mola also uses the "Blood of Kali" to brainwash local politicians and royalty to become devout servants of Kahli, does the same to Jones, and nearly has him sacrifice his ally Willie. Mola Ram, who also would kill his own men to stop Jones, is possibly the most evil character in the whole Indiana Jones franchise.
Creepy Awesome: Mola Ram manages to be one of the most memorable villains of the series even among harder to please fans of the series for how insanely dark he is with his infamous heart rip scene being a significant example of this. Considering his competition includes Nazis, this isn't an easy feat to accomplish.
Critical Dissonance: It gets enough flack that its Rotten Tomatoes score of 85% might surprise some people.
Ensemble Darkhorse: Lao Che is a fairly popular minor villain for being intelligent and the added bonus of being played by the late Roy Chiao, to the point many fans requested him to be the antagonist for a fourth film.
Nausea Fuel: The banquet. Foreign Queasine taken to its outer extreme, and quite deliberately (to squick the British and encourage Indy and company to leave).
Narm: The maharajah's condemning of Thuggee during the dinner scene sounds extremely stilted and unrealistic, though considering he's working for them at the time (though unwillingly) that was probably the intention. Kind of gives the "Did I get it right?" look he gives Chattar Lal a new spin, doesn't it?
Rooting for the Empire: Willie is so annoying, part of the audience actively wants Evil Indy to hurry up and sacrifice her already!
And on a literal level, one of the major criticisms leveled at the film is how its use of the Mighty Whitey trope seems to justify Western Imperialism (which is why the Indian government wouldn't let Spielberg film in the country).
On a related note, some Indian viewers and a few others listened to Mola Ram's big speech about using the Sankara Stones power to get rid of the English and think that it's a solid plan they can get behind. Since the actor Amrish Puri is popular in Indian cinema as a really cool villain actor, they root for him by default.
The movie contains some fairly obvious blue-screen effects that look bad even by the standards of the time. The Thuggee henchmen falling into the river past Jones at the end is one of the most notable examples.
Mooks fall into crocodile-infested water and... cut to crocodiles eating empty clothes with scream sound effects dubbed over.
Unfortunate Implications: This article goes on about how hilariously wrong the film's depiction of India is, noting that it's the kind of movie nobody could get away making in the 21st Century.
What Do You Mean, It's for Kids?: The controversy over how dark and bloody it was for a PG film partially inspired the creation of the PG-13 movie rating in the USA and the 12 in the UK.