I think it is necessary to differentiate between the two different meanings of “tuning” that can be confusing. By different tunings that could mean either different temperaments, which is what I think you are referring to, and tuning strings to actual different notes.

Temperament basically refers to the relationship of the notes to each other. Equal Temperament is what is commonly used and it basically adjusts each not so that the octaves are all in tune with each other. In other words the “space” between the notes are adjusted to effect the relationships of the notes over larger spans. One of the benefits of equal temperament being that you can transpose a piece into different keys without a significant change in the “feel” of the piece.

However when people speak of different tunings they can simply mean tuning the strings to different notes. If you are speaking to a guitarist who can’t alther the temperament of his or her instrument and talk about alternate tunings they are going to assume this aspect. Tuning the strings so that they play a specific chord when strummed open is also very common (particularly for slide players). On lever harp it is not uncommon for people to tune to either C, B or Eb depending on the kind of music they want to play and the keys that music typically falls into.

So the simple answer is that for both cases, Yes, you can tune to different temperaments and different tunings.

With different temperaments, however, you can basically only make that work if you are playing diatonically. Both lever harps and pedal harps, as they are built today, are meant to sharp or flat notes in equal temperament, so if you tune to a different temperament if you engage a lever or pedal it is going to sound off.

Tuning to alternate tunings should not have a similar problem. A semi tone is a semi tone regardless if you have a string tuned to B or Bb.

I have two lever harps and one I keep tuned in C which I use primarily for playing Celtic and folk music, and the other I tune in Eb which I use more for classical and pop music. But both are tuned to equal temperament. I have played with other temperaments for early diatonic music and it works, but many people are just so accustomed to equal that even though it is a compromise it sounds “right” to many and other tunings sound odd.