Corprus disease wasn't really a disease, but a divine infection spread throughout Morrowind by Dagoth Ur and his followers. Corprus received its power from Dagoth Ur, who in turn received his power from the Heart.
.. but Dagoth Ur and the Heart of Lorkhan were both destroyed (in a physical sense), so should Corprus still technically exist as "The Divine Disease," given that the source of divine power feeding it no longer exists on Nirn?

Dagoth Ur lost his immortality when the Nerevarine destroyed the Heart, and ALMSIVI lost their power because they couldn't access the Heart for an extended period of time (and also because the Heart was destroyed in the end), so the connections that we know existed between those individuals and Lorkhan's divinity were severed when the Heart of destroyed. Shouldn't that mean Corprus lost its divinity, whatever small connection it had to begin with, as well?

The Nerevarine gained immortality through the divine power of the Corprus disease, but he (or she) also should have lost it the second he (or she) destroyed the Heart, just as Dagoth Ur did.. right?

The heart actually wasn't so much destroyed as much as the enchantments binding the heart to Red Mountain. The Tribunal and Ur (Their connection was a little different after all) had lost their power now that the heart was unbound, but I believe that the effects of Corpus remain. It was a little different then what those with the tools used, after all. Question is if it could spread afterwards. Seeing as how the blight ended, perhaps not.

I'd be willing to accept that Corprus lost its power, and people claimed that the Nerevarine just went to Akavir after the events of Morrowind because they didn't want to admit (or simply didn't know) that he/she had died doing something simple like raiding a grotto while skimping on the quality of potions of water breathing.

Those ascended sleepers and whatnot are still around afterwards. Although Sotha Sil's Last Words implies Yagrum Bagarn's mind is clearing since the unbinding of the Heart.

IIRC, The PC also retains their immunity to diseases and the like afterward, but who knows. I doubt the Corpus Stalkers and Lame Corpus (Especially them) would regress and resume normal human forms though.

The Heart Stones in Skyrim lend credence to the idea that the Heart has a lingering effect on objects in its sphere of influence. Makes me wonder if any of the Tribunal also had side effects after being cut off from the Heart.

I think the deal is that the power of the Heart of Lorkhan still remains in people when the heart is gone (doesn't Vivec say at the end of the game he still has a little bit of left-over divinity saved up?). However, you cannot get more power from it. This means that those infected by Corprus would not be able to get further mutations. Dagoth Ur and his Heartwrights die instantly because they are a special case, only remaining alive due to their special direct connection to the Heart.