Can you be in a poly relationship if the person’s partner hates the 3rd party that’s coming into the relationship
Well, to be fair, you can do anything you want. It might not be wise, or responsible, or healthy, but there are no Polyamory Police who will forcibly prevent you.
Should you? Probably not. If you mean “coming into the relationship” to be that the 3rd party is somehow dating both existing partners, one of whom hates them…then no, that doesn’t really work. Don’t date someone you hate. Don’t date someone who hates you. That’s not a relationship, it’s some sort of weird emotional hostage situation.
If you just mean Person A is dating Person B, and Person A also wants to date Person C, but Person B hates Person C, that is possible, but opens you up to plenty of drama nonetheless. In theory, if Person A kept their relationship with Person C from affecting their relationship with Person B - say, maybe, Person A was an airline pilot and their partners lived in different cities, that might work. But barring some unique circumstances, it would be pretty hard.
You’d need to think about:
Why one partner hates the new person. If that person thinks they are abusive, dangerous, or threatening, that’s something to take strongly into consideration. If one person is violent, racist, sexist, intolerant, or actively hurtful or hateful, it’s probably best not to date them. But if the “hatred” stems from some previous drama that involved other people, or simply incompatible personalities that are independently perfectly charming, that’s a different story.
How to navigate the relationship. Can the two simply not be in a room together? That can be an issue when it comes to shared friendships, parties, holidays, etc. Will one or both partners cultivate shame, guilt, secrecy, and other nastiness by refusing to hear about or support their partner’s relationship? Or do they just prefer not to have much to do with each other and are happy to maintain a cool civility when their orbits intersect?
It’s ultimately up to the people involved. Weigh the potential risks and rewards, communicate openly with everyone involved, and handle conflict like a grownup.