Alas, Poor Scrappy: Willie briefly has a near-moment, where she's almost sacrificed in a horrific way, by a brainwashed Indy no less. Unfortunately, after surviving it, she goes right back into Damsel Scrappy mode, and the audience's pity disappears.
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"
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: Mola Ram, head of the cult of Kali Ma, intends to find the Sankara Stones to bring forth the reign of Kali in a wave of slaughter. To find the stones, he has children abducted from their villages and enslaved, forcing them into grueling labor and abuse. Ram also has a habit of ritualistic sacrifice by ripping the hearts out from his victims and dipping them in lava to burn alive and screaming. Mola also uses the "Blood of Kali" to brainwash local politicians and royalty to become devout servants of Kali, doing the same to Indy and nearly having him sacrifice his own love interest Willie, and finally attempts to plunge Indy to ravenous crocodiles in the climax, dumping his own men to their deaths while attempting to hit Indy.
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 a Signature Scene for the franchise. Others however point out that the film is too hampered by its ridiculously 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 versus 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 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 Dark Horse: Lao Che is a fairly popular minor villain for being intelligent enough to outsmart Indy 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.
Nonsensical depiction of Indian culture aside, Mola Ram is considered a very memorable villain, perhaps one of the best in the series thanks to his very intimidating cult of personality and the late Amrish Puri's larger than life performance.
Lao Che, who outwits Indy into getting what he wants and kicks off the plot of the film without suffering many losses. It helps that he is played by the late great Hong Kong actor Roy Chiao.
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). It was pretty much cooked up by Steven Spielberg and George Lucas getting suggestions about what the most disgusting possible items to put on a menu could be.
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.
"Willie... WE. ARE GOING. TO DIE!", assuming it wasn't meant to be hilarious.
Romantic Plot Tumor: Indy and Willie, in the opinion of many. Certainly the opinion of Roger Ebert, whose review recommends that you get popcorn during the "nocturnal activities" scene.
Willie is so annoying, part of the audience actively roots for the Thuggee cult to kill her.
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.
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 and heavily embellished Indian death cult.
Pretty much everything to do with India or Hinduism, for that matter.
Vindicated by History: The Temple of Doom was widely regarded as the weakest of the Indiana Jones movies, with much of the criticism being its lack of connection to its predecessor and a much darker and cynical story and tone. As time went on, no part thanks to the highly controversial fourth film, it's not uncommon to find more people who love the film for those same reasons and appreciate the change in status quo and not doing a rehash of Raiders. Nowadays, it's seen as on par with the first and third movies, sometimes even better than the other two.
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. It's a milestone in cinema history (alongside the similarly too-much-for-PG-not-enough-for-R Gremlins) for this very reason.