I don't think anything will happen long-term. Russia will piss and moan but I'm sorry, this is what you get with combat operations. Not everything can go perfectly to plan and *gasp* mishaps and accidents happen and you lose people. I'm not being flippant, it's just that if they are not expecting losses and if they are also operating dangerously close to a well-armed state that they are not fully co-operating with then they are just foolish. Now I don't think the Russians are foolish. To some extent they have to piss and moan and protest but that will be it.
There are a couple of reasons why the above may not be what will happen (or even the likely scenario at all) :
1) Turkey is not a superpower, nor is it really something the US will go to nuclear war for (in fact i don't think the US - or anyone else - would go to nuclear war for anything short of defending their own country currently). It is not tied to reality to think that in this context Turkey can afford to down a russian plane due to 'defend the homeland from planes over Syria!' or whatever.
2) Turkish current gov may be better in some stuff than similar govs it had in the past (when the West was oblivious to it all) but it is very bad in other stuff, including being at least equally terrible at human rights as those before it, and being religiously oriented and feeding sentiment for a civil war there for 2 years now. So if Russia (or others) want to bring it down they can do that without using the military anyway.
3) Turkey is a major part of the illegal immigration crisis in the EU, given virtually all illegals to Greece enter through Turkey and re-enter if deported, cause Turkey actively allows and encourages them to reach the EU through Greece.
4) We are moving closer to a dangerous world war situation with each new conflict, so maybe it would be something to calm things down if the west allows Turkey to be humbled diplomatically (likely scenario in my view), yet Erdogan will react badly to this, so it can cause civil unrest in Turkey anyway.
5) Last (
and not least, but not foremost either) keep in mind that Nato can only take an active stance (in the case of non-unprovoked war, which something tells me is not the scenario anyway) if ALL members agree to it. Now it won't just be Greece, but have you checked who our arms forces minister is? Hint: he would be more than happy to let Erdogan burn to the ground, or rather with us annexing eastern Thrace as well
(by which i mean he won't agree to anything pro-Turkey OR anti-Russia, and if he leaves gov the gov falls, so that ain't happening either).
So yeah, i think that the citizens in Turkey are likely to face serious heat. Don't know how it will play out of course.