There are very few sci-fi movies or books where relativity is important and they treat it seriously. Tau Zero is one, I'm not sure about Time for the Stars, but Forever War does have FTL.
I created a list of all these works I could find a few years ago. Here is one link:
Virtually all of Larry Niven's novels that don't have FTL touch on this. Also shows up in most of Alastair Reynolds' novels, and in the Three Body Problem series.
alastair reynold's revelation space series certainly doesn't have ftl, apart from maybe [jumper clowns spoiler deleted]; the conjoiner drives use [science] to achieve 1g acceleration to get to near light speed, and time dilation is part of many of the plots.
> the conjoiner drives use [science] to achieve 1g acceleration to get to near light speed
For [science] read [magic], more or less, but yeah. In the very first stories they were Bussard ramjets, but this got retconned out (and a Bussard ramjet actually shows up in a later book as a failed experiment). In most of the books they're more or less applied magic, though.
This, incidentally, seems to be a common theme, as later discoveries tended to fall down on the side of Bussard ramjets not working (due to insufficient density etc). The last of Niven's Known Space books have some special pleading for how the pilot has to carefully direct the ramjet to get sufficient combustion volume, a detail that was never present in the old ones.
Time for the Stars has FTL at the end. (Physical FTL, that is. Telepathic FTL exists at the beginning.) But it's otherwise it's deliberately structured around the twin paradox - the main character is even a twin.
Indeed, while I remembered the importance of time dialation in Forever War , I didn't remember the FTL travel between collapsars.
I created a list of all these works I could find a few years ago. Here is one link:
https://tvtropes.org/pmwiki/discussion.php?id=l6q90mvssgscsj...
I think this is a complete list of the films and tv series: