Flush toilets, no, but you can still use the water in aqueducts to wash away waste for communal toilets. Essentially, everything would fall into a hole and then be carried away and emptied into a river. But even in Rome there was still the standard "let's throw it into the streets" kind of thing. The Wikipedia entries for Sanitation in Ancient Rome and Roman Aqueducts are full of fun tidbits.
I'd imagine that there are chamber pots for nighttime use and convenience, and then one can make use of a privy, public or private, during the day. Though it is fiction, the Song of Ice and Fire (Game of Thrones) makes use of both as well, and Martin felt the need to depict Tyrion squatting over a chamber pot one morning.
In the devs' presentation of Kirkwall, there seems to be a halfhearted attempt to show sanitation there as well; Merrill's house in the alienage has a sort of drainage system that I assume empties into the sewers (Darktown). When you consider that it is a Tevinter-built city, this is not surprising.
Back to the dwarves, the invention of aqueducts would have been essential for survival since they live underground. There are underground river systems, so the aqueducts could be used to supply an entire subterranean city with water (and their forges!).
I'm learning so much more about ancient and medieval plumbing than I ever thought to know.

Thank you for the links. I love mundane world-building stuff like this - how people cook their food, what they use for bedding, why their clothes are designed the way they are (comfort or utility? Style? I suppose in fantasyland, style always wins out). Come to think of it, I have this notion that summers in Minrathous are somewhere shy of miserable, so everybody runs around in toga-esque garments and dhoti pants. I... have no idea why.
Also, speaking of aqueducts... err, sorta... Orzammar's clearly channeling that lava somewhere. I wonder if it's just diverted so it doesn't, you know, destroy the thaig, or if it's used as an energy resource? I imagine that's where the casteless get tossed when they die, since they're not 'worthy' of the Stone.
I reckon that Kirkwall might have had impressive sanitation back when it was Emerius, but it's slowly fallen into disrepair after the slave rebellion forced the Tevinters to abandon the city. You'd imagine that a couple Ages without magical or slave maintenance would probably tend to do that to a place.
Would also hearken back to how in our world, many towns in post-Roman Britain found themselves in similar predicaments, where lack of upkeep or knowledge on how to maintain the infrastructure caused many to be abandoned in the decades following the end of Roman occupation. At least one town had problems where the drainage being neglected caused part of the town to become a mire, if I'm recalling correctly.
Makes sense to me! No wonder everybody hates Kirkwall (or else loves it for being a miserable cesspool - looking at you, Varric!).
Makes sense the Dwarves would be the ones to first invent clocks, since they are the only race in Thedas that aren't able to rely on the sun or celestial movements to ensure accurate timekeeping is maintained.
I'd say that Tevinter and Orlais are the most likely places to see them on the surface, the Imperium having them because they've been swapping tech with the Dwarves for centuries and Orlais having them imported because of how ostentatious the nobility are.
Makes you wonder whether or not the Qunari have them as well, since they tend to be very tech-savvy?
I dunno about time-keeping, but I always pegged the Qunari as being expert navigators. Partly because of the lore... it's all but stated, yeah, but it seems like they're focused on
where and the rest of Thedas is focused on
when.