Alternatively, they respect each other, but their cultural differences lead to a lot of misunderstandings and accusations of rude behavior.
Elves need to preserve nature, as it is the source of their power. Dwarves need to harvest natural resources, as iron and coal are the source of their power. Clashing intrests are what drives this war.
Zig Zagged: The dwarven and elven kingdoms have recently come to a truce, and a dwarf and elf are sent on a mission together. Old habits die hard, however, and they're soon at each other's throats. Just as they're about to kill each other, they realize that their prejudices are meaningless, and resolve to try harder to understand one another, not as elves or dwarves, but as people. As time goes on, they come to realize that they simply can't stand each other's personalities.
Averted: Elves and dwarves get along nicely, but still not perfectly.
"You know, elf, we're supposed to hate each other, but I'm just not feelin' it. Want some ale?"
" Meh. They're just trees. I can't see why the Elders are so attached to the overgrown twigs annyways. "
Discussed: "...and for the love of Crystal Dragon Jesus, do not, under any circumstances, whatsoever, put the elven archers and the dwarvern crossbowmen in the same area, or they'll spend more time brawling each other than firing at the enemy."
Conversed: "You ever notice how, in these fantasy stories, the elves and dwarves totally hate each other?"
Deconstructed: The Elf-Versus-Dwarf war goes on for hundreds of years, and leaves both sides powerless when The Legions of Hell show up.
Played For Laughs: The team's elf and dwarf are constantly trying to one-up the other in everything, culminating in the dwarf desperately trying to learn the use of a longbow and the elf reading up on mining techniques.