What do you think happens to Skyhold if the Inquisition is disbanded?

Now that its location is known, it will invariably have to belong to someone--even if only on paper. While Ferelden and Orlais could potentially lay claim to the fortress, its location right on the border could make it a politically volatile occupation. As such, I think the Chantry may lay claim to it (especially if the Inquisition becomes part of its honor guard).

That being said, there's a decent chance it will be abandoned. It's location is such that transporting goods/resources will be a laborious and costly task. For comparison, look at Therinfal Redoubt, a fortress in Ferelden that used to train Seekers. That same fortress was abandoned right before the dragon age for financial reasons. If the Chantry was either unable or unwilling to fund a Seeker fortress that is in Ferelden wilderness, then why would they fund a fortress that is located in the farthest reaches of the frostbacks?

The only reason I would see the Chantry maintaining Skyhold is if they really wanted to maintain a presence in the far south. Or as a means of preserving the Inquisition's story (though by Trespasser they already seem to be forgetting that). Particularly if Divine Victoria is on good terms with the inquisition.

I would like for Solas (or somebody on his payroll) to set up a few wards around the place to keep anybody else out, especially anything chantric.

No modern peasants allowed 

Solas would probably reclaim his fortress, either legitimately or as bait for his enemies.

Party at the Dread Wolf's - be there or be square!

Mine castle, anyone who wants it is welcome to try.