There are some new (and some older) spells and monster special abilities that require intelligence saving throws
Apart from the spells in Deeps' answer the new spells that require intelligence saves are from Tasha's Cauldron of Everything:
- Mind Sliver
- Tasha's Mind Whip
and from Fizban's Treasury of Dragons:
- Raulothim's Psychic Lance
Also in Fizban's Treasury of Dragons all the Emerald Dragons' (from wyrmling to ancient) Disorienting Breaths cause the targets to make an intelligence saving throw.
Disorienting Breath (Recharge 5–6). The dragon exhales a wave of psychic dissonance in a 15-foot cone. Each creature in that area must make a DC 12 Intelligence saving throw. On a failed save, the creature takes 17 (5d6) psychic damage, and until the end of its next turn, when the creature makes an attack roll or an ability check, it must roll a d4 and reduce the total by the number rolled. On a successful save, the creature takes half as much damage with no additional effects.
Note that the Ancient and Adult Emerald Dragons also have Distort Perceptions as a lair action.
Distort Perceptions. The dragon attempts to alter the perceptions of one creature it can see within its lair. That creature must succeed on a DC 15 Intelligence saving throw or take 22 (4d10) psychic damage and have disadvantage on saving throws until the start of its next turn.
The Adult and Ancient Moonstone Dragons also have the Compulsive Dance lair action:
Compulsive Dance. The dragon targets a creature it can see in its lair, and a merry waltz begins to play that only the target can hear. The target must succeed on a DC 15 Intelligence saving throw or dance until initiative count 20 on the following round. The dancing creature is incapacitated and must use all its movement to dance.
Draconic Shards have the Psychic Crush ability:
Psychic Crush (Recharge 5–6). The shard unleashes a pulse of psychic power. Each creature of the shard's choice in a 60-foot-radius sphere centred on it must make a DC 20 Intelligence saving throw. On a failed save, the creature takes 55 (10d10) psychic damage and is stunned until the end of its next turn. On a successful save, the creature takes half as much damage and isn't stunned.
Similarly to that theme all the gem Greatwyrms (Topaz, Sapphire, Crystal, Emerald, Amethyst) has Psychic Beam as a legendary action which requires an intelligence save (but not their breath weapons):
Psychic Beam (Costs 3 Actions). The greatwyrm emits a beam of psychic energy in a 90-foot line that is 10 feet wide. Each creature in that area must make a DC 26 Intelligence saving throw, taking 27 (5d10) psychic damage on a failed save, or half as much damage on a successful one.
Having these dragons possibly as a main theme in a campaign would be a good reason to get proficiency in intelligence saving throws.
Having the Mind Flayers and Intellect Devourers mentioned in Deeps' answer, note that there are a couple of creatures (apart from Mind Flayer variants, like Alhoons and Illithiliches) that have identical or very similar abilities to the Tentacles and Mind Blast abilities of the Mind Flayers:
- In the Monster Manual:
- Psychic Gray Ooze which is a Gray Ooze variant
- In Volo's Guide to Monsters:
- Mindwitness which also have a Psychic ray with intelligence save
- In the Icewind Dale: Rime of the Frostmaiden adventure:
- Brain in a Jar
- Gnome Ceremorph which is basically a half-gnome half-mindflayer
- Gnome Squidling
These possibly come up in Mind Flayer themed adventures anyway, so your proficiency in intelligence saves that you picked up against Mind Flayers might become more valuable.
There are a handful of other monsters with such abilities. However there is very little connection between them so it's less likely that they would play a huge enough role in an adventure to pick proficiency in intelligence saves just for them.
- Yuan-ti Nightmare Speakers in Volo's Guide to Monsters
- Hydroloths and Duergar Mind Masters in Mordenkainen's Tome of Foes
- Mind Drinker Vampires, Nightveil Specters and Precognitive Mages in Guildmasters' Guide to Ravnica
- Sirens in the Tales from the Yawning Portal adventure book
- Inquisitor of the Mind Fires in Van Richten's Guide to Ravenloft
- Creepy Dolls in Plane Shift: Innistrad (it's a semi-official content, though those dolls are really creepy...)
- Hashalaq Quoris and Kalaraq Quorisin Eberron: Rising from the Last War
- Quandrix Professor of Theory in Strixhaven: A Curriculum of Chaos
There are some setting books and adventures where a usually powerful NPC has abilities that require intelligence saving throws even though they are not Mind Flayers or any other creatures described above. These NPCs can be leaders of a cult, a country or something, the BBEG, a powerful mini-boss along the way or even friendly to the adventurers. In no specific order:
- Fenthaza in the Tomb of Annihilation
- Iggwilv the Witch Queen in The Wild Beyond the Witchlight
- Belashyrra and Dyrrn in Eberron: Rising from the Last War
- Isperia in Guildmasters' Guide to Ravnica
- Moghadam in Infernal Machine Rebuild
- Emo in the Nerds Restoring Harmony: Adventure Together (not sure how official is this)
Having said all of that, the mere fact that we can count the intelligence saving throws in the game shows how rare they are. I haven't covered any UA, and I'm not aware of any psionic abilities of this sort that became official.