In principle, it's non-vegan, but whether a vegan would hold its use ethical or not depends a bit on the situation (and possibly on the vegan).
Uses of dung
As far as I know there are two main uses for dung:
- Collecting, processing and using it as fertiliser ("Dünger" in German)
- Collecting, drying and burning it as fuel
If you use the dung, you justify the keeping of animals
Whenever you follow any of these practices, you're giving animal husbandry a further purpose and economical justification. Disposal of dung would otherwise be a (possibly costly) counterincentive, but if you make it useful, keeping animals becomes cheaper, thus effectively supporting the industry.
This is why some vegans believe that manure should not be used, promoting e.g. Vegan organic agriculture. The same should hold true for other uses of dung than fertiliser, e.g. fire fuel.
This being said, I'm not to judge about Mongolian traditions that developed in scarcity of wood. But if there were two businesses producing the same thing, and one of them burning dried dung and the other wood, I'd buy from the one burning wood for ethical reasons. (Of course if there were a third business running on renewables, I'd buy from them, but that's a different story.)
It's possibly a different matter if you'd just collect dung from freely roaming animals. You're still interfering with the course of nature if you'd do it on a large scale (e.g. trailing behind a gnu migration or similar) and robbing it of natural fertiliser. But you're not directly depriving animals of their freedom for it.