From http://www.imperial-library.info/dwemer/guide.shtml#6
Where did the Dwemer go?
They vanished into thin air, strangely enough.
The problem is, there are many varying accounts of their end. I am going to try and split this section up into to mini sections, the (sometimes conflicting) facts of the Dwemer disappearence, along with the speculations of their disappearance.
The "Facts"
The stage was Vvardenfell, during the Battle at Red Mountain. This was the culmination of the war between the Chimer and the Dwemer. Nerevar was present from the Chimer, as were Alamelexia, Vivec, and Sotha Sil. From the Dwemer there was King Dumac and Kagrenac the Shop Foremer. Dagoth Ur from the Dagoth House was there also, as were Nords and Orcs. However, it is unknown exactly what side they were on. The Five Songs of King Wulfharth claims that the Nords and the Orcs were on the side of the Dagoth House, against the Dwemer. However, it also claims that Nerevar fought alongside the Dwemer, and the other Chimer were not present, which is unlikely. In addition, The War of the First Council claims that the Dagoth House was on the side of the Dwemer, and that the Nords and the Orcs fought with them. This, however, is also suspect, since according to Vivec's account, Dagoth Ur fought alongside Nerevar. In the end, while this question is important historically, it is unimportant in regards to the Dwemer.
In some versions of the story, the Dwemer are simply slaughtered and that is their disappearance. This is highly unlikely, as the Rourken Clan, which was across the continent at the time, disappeared at the same time. And one cannot forget that the more reliable stories (Vivec's and the Ashlanders') claim that something much more profane and meaningful happened.
These versions say that the end of the Battle, Nerevar, Dagoth Ur, Kagrenac, and others are in the Heart Chamber. Dumac may have been there and killed, or disabled. In one version Kagrenac, seeing that he is in trouble, takes the tools and walks over to the Heart. The others see him attempt to use the tools on the Heart, and at that moment all of the Dwemer on Tamriel vanish without a trace. His use of the tools was probably supposed to elevate his race to something like Godhood. In other versions, Kagrenac is killed by Dagoth Ur, and Azura shows Nerevar how to use the tools to sever the Dwemer's link with the Heart. "And on the fields, the Tribunal and their armies watched as the Dwemer turned into dust all around them as their stolen immortality was taken away." The question is whether or not the tools were actually used, and whether or not they actually worked.
Speculation
Assuming that the above account of the Dwemer's End is correct, there are several possibilities as to what became of them. I will list the ones that I have thought of (and think are the most likely) here.
Possibility A: The tools worked. Kagrenac successfully used his tools, and possibly The Calling, to take his race and remove them from Mundus, the Mortal Plane. He either turned them into Deities or simply allowed them to move freely in the Outer Realms.
This is possible, but unlikely. If it had happened, then Vivec, who is a God himself, probably wouldn't have said that "I have no sense of them in the timeless divine world outside of mortal time". In addition, for reasons discussed below, I believe that certain Et'Ada would never have allowed the Dwemer to carry out this act of arrogance. Finally, one might think that if the Dwemer had become Gods, we would have heard more about it.
Possibility B: The tools failed. Kagrenac foolishly de-created his race and erased them from Nirn. They were either destroyed entirely or were transfered to somewhere outside of Nirn and died.
This is more likely. The book "The Egg of Time" seems to indicate that there was a debate going on. The book argues that it is safe to tap the power of the Heart, and that the Dwemer who believed that it was a great risk were incorrect. I believe that this kind of "debate" wouldn't have been revealed to us unless tampering with the Heart actually was dangerous, and that something terrible very well may have happened if Kagrenac had tried to use the tools on it. Also, there exists an article entitled Sermon Zero of the Thirty-and-Six-and-Nine Sermons, which was given to us by Jobasha on the official forums. It is highly confusing and likely not considered an official document, but it contains three secret messages which can be deciphered from the text. One of them reads "To the Dwemer and Oblivion belong this treasure and they are there dead".
Possibility C: Kagrenac had no chance to actually use the tools. The split second that he did, Daedra Prince Azura snatched him and all of the other Dwemer from Nirn, and imprisoned them in Oblivion. Alternatively, he did use the tools, but as soon as his race was removed from Mundus, Azura imprisoned them in Oblivion. Or he died before he had the chance, and Azura instructed Nerevar to use the tools to destroy the Dwemer enchantment upon the Heart, making them instantly disappear. Either way, Azura put them in Oblivion and there they may or may not have died.
This, in my opinion, is the most likely of all of the possibilities. As has been discussed elsewhere, the Dwemer doubted the power of the Et'Ada and did not venerate them as the other races did. One specific example of this directly involves Azura- the account contained in Azura and the Box. And one cannot forget the very idea of Numidium, The Brass God. The construct which, according to Divine Metaphysics, was a new God constructed from the power of a "Dead" God, Lorkhan. This is blasphemy in the highest sense of the word, and according to several accounts, Azura did not want it completed.
More importantly, Azura was involved several times in the affairs at Battle of Red Mountain, before and after the war. In The Battle of Red Mountain, it is said that Nerevar went to Holomayan and summoned Azura to receive counsel, and there she told him that the construction of the "New God of the Dwemer should be prevented at all costs". In Nerevar at Red Mountain, Azura instructs Nerevar while in the Heart Chamber to use the tools on the Heart to destroy the Dwemer, which he does (at this moment, according to the legend, the Dwemer simply all vanish). Later, according to both accounts, the Tribunal break their oath to Nerevar and Azura and use the Heart to become Gods. At this time Azura appears and curses the Chimer, turning them into Dunmer "that they might know forever their wicked deeds ". And finally, it might be said that Azura returned her champion Nerevar to Morrowind to once and for all destroy the Heart and its destructive influence, and that this is the story of the Nerevarine.
All of these events show that Azura was an active force in the prevention of the use of the Heart. For whatever reason, she did not want the Heart to be used for profane purposes. Thus, I find it likely that Azura, whether it be to protect some kind of natural order regarding divinity and mortality, to protect the safety of those on Tamriel, or simply as a punishment for the Dwemer arrogance and hubris, imprisoned the Dwemer in Oblivion forever, and there they still exist.