It established that time dilation doesn't affect ships at warp. (Do ships at warp experience time dilation?)
And in this answer to this question
@chepner points out that full impulse is 1/4 of the speed of light, which should equate to about a 3% time dilation effect.
In a word? No.
Star Trek never denies Relativity, and the resulting time dilation effects, but it never really deals with them. Even ships traveling at impulse (and hence, sub-light speeds) are never seen to have to deal with relativistic effects.
The five series and ten movies are incredibly inconsistent in their use of impulse power fractions and the time it actually takes to get somewhere. However, according to Memory Alpha's article on the subject, there's at least one reference out there that suggests that full impulse power is roughly equivalent to 0.25c. That velocity ought to be fast enough to involve a noticeable time dilation effect (0.10c is generally thought of as the threshold beyond which it's a real factor, although it's measurable with atomic clocks even today on the International Space Station!).
However, it's worth noting that time dilation only really matters if you have a ship that's traveling for a long period of time at a significant fraction of light, which most of the ships we see in Star Trek never do. For example, Julian Bashir suggested it would take "seventeen years, two months, and three days [...] give or take an hour." (DS9: A Time To Stand) for them to return to Federation space at sublight speeds when the warp drive on their Jem'Hadar ship is fried. That trip would undoubtedly have involved time dilation relative to their destination (Starbase 375), but we never find out because they never actually do it.
None of this means that time dilation is somehow not an issue in the Star Trek universe, so much as that it's never been an issue raised within the context of a story. The Stardate system was originally developed by Roddenberry and Sam Peeples (writer for "Where No Man Has Gone Before") to side-step the question of intragalactic time keeping. As originally envisioned, the only thing that mattered was that the stardates in a given script were internally consistent. As far as the show-runners of the Original Series were concerned, the stardate could in fact be different in different places, thus implicitly allowing for time dilation and relativity (and also the fact that they hadn't really decided how warp drive worked and how much it negated such things), while not really having to address it. (Memory Alpha: Stardate) Everyone just accepts that, as a starfaring culture, time is relative, and dates, even stardates, are conventions for counting local time.
The later series' passion for nailing things down to a real timeline relative to our current Earth calendar fudged this intention, but Star Trek retained the basic idea that space travelers just cope with time dilation if it happens, and never really address it otherwise.
Yes. In the Star Trek: The Next Generation episode "The Price" (season 3, episode 8) a shuttlecraft piloted by Data enters a wormhole. Communications with the shuttlecraft from the Enterprise are affected by time dilation experienced by the shuttle.
DATA'S COM VOICE: Captain, we have penetrated the outer event horizon of the wormhole. Sensors read severe doppler shifts...
(distorted)What the bridge crew is hearing is gibberish. Picard frowns.
WESLEY: It's the relativistic time dilation, sir. Time is moving more slowly inside their part of the wormhole. I can process the audio to compensate...
Actually, it was first done in TOS: Tomorrow Is Yesterday. There's an article on Memory Alpha about the Slingshot Effect. It says:
In 2267, the Starship USS Enterprise accidentally traveled through time to the late 1960s, when an encounter with a previously-uncharted black star required the crew to utilize all warp power in reverse to break away from the star's powerful gravitational attraction, creating a whiplash effect. In order to return to their own time, the crew of the Enterprise recreated the accident, using the gravitational pull of Earth's sun to perform the slingshot.
I think this is what you're thinking of- from what I can see, Warp Fields compensate for relativistic effects by compressing the space in front and lengthening it behind, so temporal effects are normally compensated for.
As for whether impulse speed causes relativistic effects, that's never shown. I'm not sure, but I think that impulse still generates a weak warp field, thus negating such effects. I'm not aware that there's any form of sub-light shown that doesn't involve either propellant or warp drive.
There is, of course, the question of whether these are relativistic effects as we know them given that they were already going faster than light before the slingshot. If you're asking whether pure speed can cause it in Star Trek, the answer is yes. But whether it's Einsteinian relativity, that's a little harder to answer.
There is the episode where Scotty implodes the warp engines in a cold start babble babble babble, to save the Enterprise from burning up in the atmosphere. They end up going "faster than is possible for normal space", i.e. faster than the speed of light without entering warp.
They had leapt over the infinite energy required at lightspeed to faster-than-light travel. Per relativity, they started going backwards in time.
So it answers the original question in spades, without delving into warp and slingshot (a silly concept that should never have been introduced IMO.)
cvessel that turned out to be a pair of researchers who left slightly before warp was discovered and who were rather disappointed at having lost so much time in vain