Elves and dwarves tend to be strong in their positions of power: The Dwarves in their mountain holds, and Elves in their forests, which are protected with magical barriers. it's hard to imagine a threat which could threaten both. Usually what undoes them is when they wake up something to which those barriers represent no obstacle, such as the balrog. Armies of Orks or Goblins are really only a threat when someone coordinates them, as they will end up killing each other
I'm not all that familiar with middle earth mythology but I guess any powerful mortal or semi-divine being could be a real threat...
Actually.. how about something like this... What if a mortal man, a noble lord of (your setting) has come by a magical artifact which promises great power, but at a price. I'm not sure what this artefact could be but it could allow him to circumvent the early warning systems of the Dwarves and Elves.. what if it is using him to get each of them to destroy each other.. I.e. he attacks the elves, and none survive, leaving behind only a broken Dwarven axe, then he attacks the Dwarves, leaving only elven arrows in the dwarven defenders... Then lets the carnage ensue. The Dwarves and Elves carry over grudges since the 1st age, so it wouldn't be hard to fan those flames!
I did consider it might actually be one of the Dwarven rings.. http://www.glyphweb.com/ARDA/r/ring.html seems to indicatre they were destroyed, but certainly in the film monologue, it said the Dwarves 'lost' them. It's hard to know what they would or could do once the one ring was destroyed, but they were powerful artefacts BEFORE it was made, so they would likelly be afterwards, just as the elven rings were! Another interesting thing I have read is that narsil was actually forged by a Dwarf. The Dwarves themselves were powerful forgers, and who knows what they may have made?!