This answer will mostly be opinion based, but I'll use some canonical sources to try and support my arguments and answers.
Firstly, I highly doubt Dwarves produced waste other than food waste due to the lifestyle they lead.
I don't think there would be much waste created by the Dwarves. They likely don't waste anything they eat, this could be because they have big appetites or because they simply dislike wasting. If any waste food is created they may be able to trade back with the Men they originally traded with for food:
Men became the chief providers of food, as herdsmen, shepherds, and land-tillers, which the Dwarves exchanged for work as builders, roadmakers, miners, and the makers of things of craft, from useful tools to weapons and arms and many other things of great cost and skill.
The History of Middle Earth: The People's of Middle Earth - Of Men and Dwarves
The Men may have found out that food waste could be used as fertilisers and would therefore be "of great cost" and the Dwarves could therefore trade that for other food.
Furthermore, one of the Dwarves in the First Age (Mîm) talked about eating roots, which could be cooked as well as hoarded "like the nuts of a squirrel"
But when they were cooked these roots proved good to eat, somewhat like bread; and the outlaws were glad of them, for they had long lacked bread save when they could steal it. ‘Wild Elves know them not; Grey-elves have not found them; the proud ones from over the Sea are too proud to delve,’ said Mîm.
‘What is their name?’ said Turin.
Mîm looked at him sidelong. ‘They have no name, save in the dwarf-tongue, which we do not teach,’ he said. ‘And we do not teach Men to find them, for Men are greedy and thriftless, and would not spare till all the plants had perished; whereas now they pass them by as they go blundering in the wild. No more will you learn of me; but you may have enough of my bounty, as long as you speak fair and do not spy or steal.’ Then again he laughed in his throat. ‘They are of great worth,’ he said. ‘More than gold in the hungry winter, for they may be hoarded like the nuts of a squirrel, and already we were building our store from the first that are ripe. But you are fools, if you think that I would not be parted from one small load even for the saving of my life.’
The Children of Hurin: Chapter VII - Of Mîm the Dwarf
Mîm's roots may mean that dwarves didn't have to waste food as they could cook as many roots as they needed to eat. Although as isanae pointed out, Mîm is what's known as a "Petty-Dwarf", which "were banished in ancient days from the great Dwarf-cities of the east". Although, this may not make them a good representation, their name in the dwarf-tongue is known and therefore makes it likely he learnt of them when he still lived (or his ancestors) in the "great Dwarf-cities" of old.
Dwarves show an aptitude for building tunnels or shafts to the outside. This would suggest the possibility of having shafts out of the mountain which would be used for body waste removal (and possibly other/food waste as well).
There used to be great windows on the mountain-side, and shafts leading out to the light in the upper reaches of the Mines.
huddled close together in a corner to escape the draught: there seemed to be a steady inflow of chill air through the eastern archway.
The Lord of the Rings: The Fellowship of the Ring, Chapter IV - A Journey in the Dark
Other options would include waterways or simply pumping out the waste with hand-pumps or Archimedes screws. Dwarves may have possibly used aqueducts and leats to provide flowing water to their mansions, and be used as sewage. They may also have used pumps similar to that of the Romans (below) to pump waste water out of the mountain, possibly having waste reservoirs to pump to, until having either pumped out of the mountain or up to the surface (Bearing in mind that simply because they were underground didn't mean they couldn't be at a higher altitude to the base of the mountain).