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<small>Disclaimer: I am no lawyer or otherwise law expert, all the following is to the best of my knowledge and does not constitute legal advise; when in doubt, as a lawyer.</small>

There are several things to consider.

First, the general idea. In the case of Warp drives, the general idea would be: A machine that allows faster-than-light travel by manipulating (warping) the spacetime around it.

The general idea is definitely not protected in any way. The only legal tool to protect ideas are patents, but those only come into play if you actually want to build something. That is, even if there were a patent on Warp drives (I doubt it, but who knows), it would not stop you from using similar technology in your stories (indeed I think you could even base your fictional device closely on the description in such a patent; but about that I'm not completely sure, because the text of the patent is probably copyright-protected as well).

Next, the detailed workings of the machine. For example, for Star Trek's Warp drive, this would be to produce a Warp field that causes a subspace bubble which then distorts spacetime. That's concrete enough that I'd expect it to be protected, but even if not, it is certainly concrete enough that you might face litigation. And even litigation that you win is generally worse than no litigation at all. Remove the subspace part, and I think it might already be general enough.

Note that basing your description on actual science should be a fair game, since facts are not copyright-protected. So e.g. basing your Warp drive replacement on the hypothetical Alcubierre drive should be safe (of course copying the wording of any article about it would not be OK, but you'd not want to do that anyway).

Next, the name. Here not only copyright has to be considered, but also trademarks. For example, the term Star Trek is trademarked, so even after the first Star Trek series goes out of copyright, you still cannot freely use that name. This applies even if your content is in no other way related to the Star Trek universe (e.g. say you are writing about a trek of Hollywood stars).

About the name Warp drive, I don't think copyright has anything to say about it (it's basically a direct description of what the drive is supposed to do, which is to warp spacetime). However given that the name is generally associated with the Star Trek franchise, it *might* be trademarked. Then again, other answers mentioned the use of the term in non-Star Trek contexts (including earlier ones), so it may not be trademarked. But then, AFAIK earlier use of the term by others is no guarantee of non-trademark; only if the term was in common use, this would prevent the trademark. If you used the term before the trademark was established, I think you are allowed to continue using it, but that doesn't give others the right to use it, too.

An important part of it is that you lose a trademark if you don't enforce it; this means that companies are likely to do so even where the chances of winning in court are slim (especially given that there's a good chance that the other side will give in before the case even goes to court). Again, you probably want to avoid litigation even when your chances to win are high, so the best bet would be to avoid such names, especially given that it is easy to give your own names to it.

The simplest possibility is to just use the generic term FTL drive. Or you could name it after whoever invented it in your universe (just like the Diesel engine is named after Rudolf Diesel or the Edison phonograph is named after Thomas Edison), Or you could drive it after the fictional fuel is consumes (like the petrol engine/gasoline engine), or something else that is used in its mechanism (maybe it makes use of tachyons to control the spacetime bubble, then it could be called tachyon drive).