he's worried that things might get really awkward
If things go bad, it might damage the friendship the two have for each other
I'm gonna throw in my experience here as a cautionary tale for you on this one. I had a relationship of 4 years be crushed, twisted and killed off over another 1 1/2 years because we failed to manage this scenario properly. My partner Grotto and I had not discussed clear boundaries around me hooking up with or dating his friends. Grotto had, though, like your boyfriend, voiced to me fears that things could get messy between him and his friends if I had an intimate relationship with one of his friends. We agreed to discuss this more at some point, but never got around to it.
Some time later, I had unplanned sex with one of his friends, Lobe. I had not discussed it with Grotto beforehand. The immediate fallout was bad. Without going into too many details, though, this eventually became a negotiated okay by Grotto (for me to date Lobe). There was a lot of messed up emotions we were regularly wading through, though. I think a major factor was Grotto and me not unpacking Grotto's boundaries around this, properly, in advance. In retrospect, Grotto regretted saying yes to this. It got very very ugly. I would have said Grotto and I had a healthy relationship and good communication skills - before this. We had been through many challenges, but this broke us.
Ultimately Grotto and I broke up. 3 years later it's still sad and raw. Grotto and Lobe don't talk anymore (they had been close friends for decades) and the fallout broke their friendship group. Grotto is still suffering from this as he has not been able to find a healthy way through the pain. I know my contribution is just one part of the story, but from a certain angle, my decisions and carelessness caused incredible ongoing pain to someone I love and care for (Grotto), and toppled more than one 'lifelong' friendship.
I wish I had talked more with Grotto about his concerns regarding me dating his friends. I wish we had explored the possibility that he had a hard boundary around this. I wish we had not relied upon our laissez-faire and good faith attitudes towards doing poly. Sometimes, meaning well is not enough. Sometimes you can fuck up bad and hurt people deeply without intending to.
I value friendships highly and I risked the great friendship between Grotto and Lobe, as well as other relationships in their friends group, on top of risking my individual relationships with both Grotto and Lobe.
So, I agree with Kitkatharsis that the worst that could happen is their friendship and your own romantic connections ending. I just wanted to add my own visceral spin to it.
If I had a do-over, I'd do it differently.