Stargate SG-1, Season 4, Episode 6 has a variant of the loop trope, but everyone (including most of the protagonists, and everyone else on earth) don't remember what happens, while two protagonists remember every loop until they are able to stop the looping.

They debrief the others who don't remember at the end (except for the things they did when they took a loop off anyway!) - but they didn't miss too much since everyone else on earth missed it.

Another fictional work - a book, not a movie / TV show / anime - is Stephen Fry's 1996 novel Making History. The time travel aspect is questionable - he sends things back in time to stop Hitler being born, but no people travel through time. However, he remembers the past before his change, and has to deal with the consequences of having the wrong memories relative to everyone else.

12 Monkeys

12 Monkeys with Bruce Willis. Not exactly amnesia but extreme confusion and lack of emotional control. Some of that was due to time travel and some due to the extreme emotional trauma his character experienced.

There was a series on SyFy around 2015 called Dark Matter about a crew that wakes up on a spaceship with no memories. It was okay.

I'm pretty sure there's an anime of a very similar premise on Netflix called "Erased".
It reminded me a lot of the movie "butterfly effect"

Steins;Gate and its sequel movie Steins;Gate: The Movie − Load Region of Déjà Vu

Weirdly, season 4 of both Fringe and Eureka have a portion of the main cast shunted into an altered timeline and having to reconcile their original memories with their "new" histories, to varying degrees of success.

Travelers kinda inverts the premise in its second season, where a bunch of time travellers sent back to fix the past start seeing their superior foreknowledge slowly rendered useless by the fact that their mission is actually succeeding in changing the future.

There might be an All My Circuits episode in that vein https://www.youtube.com/watch?v=79v6wWIoY18

Looper. It's a bad movie, but there's a struggle with memory for the people from future.

There is just way to much Isekai about this. Just search some time related tag on any manga site or smth and you'll find more than you can ever read.

Quantum Leap. The original series with Scott Bakula was much better.

Project Hail Mary matches this, will be a movie soon

Not exactly the same premise that you described, but just a first reaction on the title is try Final Fantasy X. It’s certainly got time travel elements as well as amnesia!

Genuine Answer: Predestination

https://en.m.wikipedia.org/wiki/Predestination_(film)

Could be a partial spoiler but Date a Live has it involved in the general story. If I state how it'll work it would be a major spoiler, so not detailing it further.

So the premise is: they travel back in time, change their future, travel back to the present, forget everything they did while time traveling, and now have to just experience their new future without knowing anything changed? Do I have a that right?

I mean, how do you know this isn't the backstory for every character in every story ever? How do you know you didn't JUST do this yourself?