I'm trying to find a translation for this phrase, with the meaning of "don't let up" to be similar to "don't stop," or "don't give up," or "don't give in." Building a motto for a logo. I've looked through the site as seen my best bet for the "don't" part is to use either "Noli..." or "Ne...", so part of my question would include which of those is appropriate. I just don't know something similar to "let up" that would work and not come off as gibberish.
Noli would not be appropriate, because negative requests with noli are pointedly polite. An exhortation to keep at it and not give up is direct and to the point, not polite; just like you don't say: “May I ask that you not give up” in English either.
So ne it is. The following verb should be in the perfect subjunctive.
(Another option would be cave, but that would be closer to saying “take care that you don't give up, by no means give up,” etc. A nice discussion of the various options you have can be found here.)
For the rest of your phrase, I would suggest:
Ne hastas abieceris.
Literally: don't throw away the lances. Hastas abicere is a figure of speech (I.A.2) meaning “to give up, to lose courage.” I do not think it was a common Roman expression, but Cicero certainly used it in his speech pro Murena (45).
The following words all have the sense of persevering or pushing forwards:
So any of these can be used. The word deficio mean to give up. So you can say:
Which means "Don't giveup." The Romans often used the word cedo in this context, so we have:
Cede nullis. (Yield to nothing)
Ne cede. (Do not give up)