Americans Hate Tingle: Even though the movie was a box-office smash, the government of India were not pleased with the negative depictions of their country, and their national censors temporarily banned it from being shown in India.
The Indian government found the script so offensive, with absolutely ridiculous baby snakes, eyeball soup, beetles, and chilled monkey brains being served at the banquet, and Kali being misrepresented as a goddess of the underworld instead of a goddess of change and empowerment; the government denied filmmakers permission to shoot in India, with Sri Lanka being used as an alternative for the scenes that took place in India, which only made things worse with its Orientalism.
In general, the Indiana Jones films were never well known and popular to begin with in India, so it's unlikely the film would have done well despite these issues of representation. Indians generally didn't quite get the B-Movie serial nostalgia on which the series is builtnote Star Wars is another series that isn't quite widely liked in India for the same reasons. Indian audiences generally prefer the less fantastic and realistic Hollywood action films and genres, like The Terminator or Jurassic Park since those genres and its tropes never quite faded in India. Spielberg himself remains considerably popular in India, with his later films being quite successful, influential and well known, and playing without fuss with the Indian censors.
Awesome Music: The very catchy "Slave Children's Crusade" or the majestic "Short Round's Theme"
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.
Contested Sequel: Was it an interesting variation on the plot of Raiders, or was it too dark and gory to enjoy? It is generally seen as the weakest of the original trilogy, but reception of this movie warmed a bit among the fandom after the release of the even more divisive Indiana Jones and the Kingdom of the Crystal Skull who appreciate its weirdness and its attempt to be different from the first film and that it has many of the franchise's Signature Scene. Others however point out that the film is too hampered by its racist and inaccurate depiction of Indian culture and its indulgence of Mighty Whitey tropes to truly rise to any level of greatness. The presence of Willie and Short Round vs. a returning Marion Ravenwood and Mutt Williams doesn't help things, either.
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.
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?
Some of the faces Harrison Ford pulls make certain moments very hard to take seriously, especially around the bridge scene. "Mola Ram, prepare to meet Kali... in Hell!", a rather silly line in its own right, is delivered with enough raw ham to feed every Thuggee in the next three square miles.
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. And the cherry on top is that they didn't even get crocodiles, obviously using American alligators as stand ins.
Squick: The infamous dinner sequence. Complete with servings of snakes filled with eels, eyeball soup, beetles, and chilled monkey brains for dessert. The chamber full of bugs also qualifies.
Values Dissonance: Indy forcefully grabbing Willie with his whip for a kiss without notice after an argument is viewed as very blatant sexual harassment now. Not to mention the overt Mighty Whitey themes of the story all about the heroic American superman going into India and fighting the evil totally made-up) Indian cannibal cult.
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.