There are a few different issues. Some of the techniques they used for Star Wars with compositing the visual effects (putting together the different layers of film into one image) were meant to be viewed through a projector on a screen, which reacts to brightness and darkness in the image differently than a TV screen, which displays the image directly. The big issues were "matte boxes," rough outlines of objects like spaceships that cut out the stage lights and backgrounds from the film and just left the model and the bluescreen behind it, but the bluescreen wasn't entirely eliminated by their process, so there'd be a sort of rectangular outline around different objects. There were some other invisible clean-up shots like that in the special editions (one I remember is that the hallway to Leia's cell on the Death Star had a painted background that showed it extending forever, but there were shots where you could see it didn't line up with the real hallway, so they replaced it with CGI for all of them).
What they do in cases like those is going to be a matter of judgement and taste for the people doing the restoration. I would guess things like replacing the painted backdrop with a nearly identical CG one is probably going to be reverted back to how it was in 1977, but the matte boxes might be removed or obscured as part of the clean-up and color-correction process any re-release would go through.
In TOS, the issue wasn't that the special effects weren't in HD, the show was edited on film, so the base quality was just as good as the live-action footage, it's that the techniques they were using meant that the quality of the filmed special effects elements went down substantially when they removed the blue screen behind the model and put in the stars, and the shots were reused over and over and became damaged and degraded, so the same stock shot in late season 3 looked much worse than it had in early season1. That wasn't a big deal broadcasting on TV over the air, where the resolution was low and there was going to be some amount of static, but even though the footage was technically in HD, it looked noticably worse than the live-action stuff in a way it hadn't when it was all being seen as a standard-definition TV broadcast.
