This falls under the category of “no fun”; having two partners that you care about who don’t get along, or one doesn’t like another, etc.
There are several aspects of this worth discussion:
Prevention: One way to go, of course, is to try not to get into this situation to begin with. That can sometimes be easy. If an existing partner doesn’t like someone you’re interested, and you’re not THAT interested, then letting it go is not so bad.
Sometimes, of course, that’s harder. You may be so attracted to the person that it makes you really unhappy to let it go. Or, you may not *know* that they don’t get along; they may not have spent enough time together to tell.
I’m a fan of avoiding “no fun” situations, so I go with prevention when I can. That means (outside of “Festival Rules”) getting to know someone socially, with my partners, as part of leading up to dating/getting involved. Even a first impression can help a lot.
Jealousy: Sometimes it’s tricky to separate out your existing partner not liking someone new from hir possible jealousy of that person. I have no brilliant advice on this one, except, as usual, “Talk it out.” Also keep in mind that some level of irritation with a new person is pretty normal; NRE will cause YOU to gloss over all sorts of annoying little things, but your partner is not also in NRE with this new person.
Separation: One obvious strategy for dealing with partners who don’t get along is to limit the amount of time they have to spend with each other. That might also mean watching how you talk about one to the other; reminding hir of the things that annoy hir will almost certainly not help.
Reconciliation: Another strategy is to see if somehow the relationship between the two of them can be improved. that might mean addressing the issues that are bugging them head on, either with your support or with you locked out of the room. 🙂 It also might mean giving those issues less weight, by finding other things that they can do together or interests they have in common.