There is always a ready supply of slaves because the other nations, whilst officially outlawing the practice in their own states, aren't above selling their citizens into slavery. I would imagine that what Loghain did is more common than you might imagine. Also there are the activities of pirates, who attack coastal locations and take the inhabitants prisoner to sell off as slave, plus slave hunters on the mainland, particularly in the regions that border Tevinter, who likely prey on isolated communities.
As for natural resources in Tevinter, I would imagine there are plenty of quarries and mines where they can extract the more mundane resources they need. The slave workforce would of course be used for this. I would think the inland cities of Marothius, Perivantium and Solas are likely involved in mining and the like as they would be in the rain shadow of the mountains and thus likely too dry for agriculture. There are probably numerous smaller settlements across the Imperium that are likewise engaged.
They likely have more agriculture nearer the coast and in the north, where the climate will at the very least be Mediterranean but around Minrathous possibly even sub-tropical. They probably have a sizeable fishing fleet. Since Dorian even has silk undergarments, I would think that Tevinter probably has its own silk industry and this would be another source of trade with other nations. All this will be run with slaves, which is why their economy would collapse if slavery was abolished overnight.
The only resources likely not dependent on slaves would be magical items, the lesser ones of which they may trade with other nations. Other than that I would think they would concentrate on producing things that are valued by the dwarves, since they rely on them for lyrium.