If you're having fun I fail to see what the issue is? I played with a similar situation when my elf hated orcs for burning his villiage to the ground, then one of the other players came with a half-orc PC. My PC would routinely insult the other's intelligence and we had competitions for kills (my rogue usually out-killed his PC) so I could rub it in even harder. We were both nobles though so neither could attack the other without our families coming to blows and likely execution. One of my favourite campaigns ever because it's memorable.
If eladrin and drow hate each other you and your friend made PC's knowing they'd not get along so it seems like it was inevitable. Now, your PC's can come to realize that the preconceived notions they had of the other race were just that, prejudices, and you can have an "After school special" moment where you come to see them as the people they are rather than by the colour of their skin.