A theoretical victory is good... But it isn't enough for the Ottomans. The Russians have too many troops there with some of their best commanders. My expectation is the Ottomans Army would stay on the defense in Sarikamish until Spring. If the situation allows it (Russian counteroffensive failing miserably) then the Third Army (if the losses are not large) make one more move to Kars until stopping most offensives. Kars helps the Ottoman Army to put a better defensive position in Anatolia, which is necessary. The Third Army cannot drive the Russians out of the Southern Caucasus, but only take a good position to halt Russian advances to Eastern Anatolia. The Armenian Groups are in a tough spot. If they rebel, they will be taken care with a harsh hand, unfortunately with a part of a large relocation attempt to Syria as OTL. I wouldn't expect it though some kind of sabotage attempts will be made.
Between 1915-1917 a lot of things can happen. If the Third Army can keep Sarikamish/Kars until the Russians lose more against the Germans in Poland and the Baltics, they will be forced to send more troops in that direction considering the Ottoman Army cannot really take offensive attempts against the Russians. When the Russians are pretty much broken and in revolution, the Ottoman Army start their final assault on the area of Yerevan and Gyumri as the resistance is gone.
If the Russians beat the Third Army in the period 1915-1917, the situation can be like OTL or only a little worse. With regards to the Armenian situation it may not be too different if the Armenians still start an insurgency so close to the Front.
The consequences are that no large Armenian refugees move to Russian Armenia. Demographics are not largely changed. The Ottoman Army will try to retake Batum at least, which is guaranteed. If the Ottoman Government cannot secure German support to annex Azerbaijan and Armenia then they will support an Azerbaijani Movement for independence, hoping to annex it in the near future.
Militarily speaking, the Eastern Front offers opportunities. When the Gallipolli Front has ended, the Ottoman Government can send additional troops to the Eastern Front to reinforce it. So the Third Army needs to hold Sarikamish/Kars until mid 1916, reinforcements can finally arrive. This also depends on the situation as Istanbul will also consider moving troops to Iraq or Palestine before things get overheated there. Holding Sarikamish/Kars has an opportunity as the Ottoman Army have more men at one place rather than less men in two places (Erzurum-Bitlis) for defensive purposes. That is the only thing the Ottoman Army needs to do. The only thing that they can do considering the resources of the Ottoman Empire and the circumstances.
Consequences for Russia is hoping to get a quick victory against the Ottoman Army after such defeat, which could lead to more rash decisions. But then again, some of their finest commanders were there. If the news of defeats and failed attempts to flush out the Ottoman Army out of Kars leads to nothing, the Czar could end up even more in trouble. In short, Russia won't necessarily lose most of the Souther Caucasus apart of the Kars Oblast (Minus Batum) but attempts to retake it and the rising casualties can put it in bigger trouble than OTL. A Russian Revolution in November-December 1916? Who knows... Another thing... The loss of two battles in a short time in the Caucasus means Russia will likely retreat troops from Persia to reinforce the losses, rather than stripping troops from the Eastern European Front or conscripting more peasants to start an offensive again. If Enver plays his cards right he can stay on the defensive and provoke a rebellion in Persia that the Russians and the British have to deal with resulting in more casualties. An even less unlikely but not impossible thing is Enver trying to lure the Qajar Shah in the war when the British and the Russians are weakened in Persia by helping them to recover most of Azerbaijan and Armenia. Not that Ahmad Shah Qajar would go for it but Enver might consider it...
But how the battle ends in a victory... that is important. The soldiers were not prepared for Winter and did not really have enough numbers to secure a victory without too much casualties. If the Russians somehow miscommunicate which leads to panic in the ranks and a route again, this could be ideal.