Could a Dwarf melee with a Balrog?

[The Might]

It sure can in LotR: Conquest, fighting Gimli costed me almost half of my hitpoints as a balrog in Moria.

No, but seriously now, I guess that if Elves could find them then well armoured Dwarves could too. Of course probably not with the same efficiency, but still it would be possible.

[Hookbill the Goomba]

Alone? Probably not. Most of those who fought Balrogs died. Even the ones who won.

I guess range would be the crucial issue.
The Balrog of Moria is described as being "of man-shape maybe, yet greater". So the Balrog would have all the time in the world to hit the Dwarf with its whip or sword before the Dwarf could get close enough to make a dent.

Not sure about this. As you said above, in The Battle Of Unnumbered Tears, the Dwarves managed to fend off Glaurung. Granted, he was not yet at his full stature, but that's still an achievement. If in a big enough group, I'd say the Dwarves could give a Balrog a run for its money. But then, Balrogs are probably mightier than Dragons. I present you with the coolest thing in all of Tolkien;

Originally Posted by Book of Lost Tales II, Chapter III, The Fall of Gondolin

Now then the plan that they made was to hold what they had won, while those serpants of bronze and with great feet for trampling climbed slowly over those of iron, and reaching the walls there opened a breach where-through the Balrogs might ride upon the dragons of flame;[...]

Although, if we're talking a steed-rider battle, I wouldn't fancy myself beating a horse in a fight to the death.