This is one of the age old relationship dilemmas:  best friends of the opposite sex.  I'm sorry to say that you're going to have to choose.  As much as I hate choosing one friend over the other on principle, relationships don't live in that vacuum.  Relationships are filled with a lot of emotional and irrational give and take to comfort and provide solace to your significant other.

This is one of those situations where that give-and-take requires you to do something you really don't want to do for the sake of your boyfriends peace-of-mind.  Ultimately that's the goal of any person in a relationship - to provide the other person some peace-of-mind.  If he's uncomfortable, well, he's your boyfriend and his opinion trumps all unless he's being completely unreasonable.  And in this case, he isn't.  Perhaps he doesn't understand the relationship because he doesn't believe that men and women cannot just be friends without some ulterior motives.  Right or wrong, that's how he feels and you have to respect that.  I'm guessing he told you that at some point and you chose to ignore it or just avoid the inevitable.

This is your time to evaluate your relationship.  Your boyfriend is requesting you to let your best friend go so you have to decide who's more important.  Do you see a future with this boyfriend beyond the current situation? Is it worth it to leave the best friend behind?  If you don't see a future, then the choice might not be that hard.  But this will come up in every relationship you have, so you might as well get prepared for that.

Or hope that your best friend is gay.