SL is made of many cultures and crosses all the borders that will become the front line-personally, what me and the people I interact do is that we dont share these type of information about our Real Life. Theres a saying, "ignorance is bliss" and sometimes, its just better to not know something. In order to aviod political tension between friends and those you deal with, simply dont tell anyone where you are from and who's side your own, your beliefs, etc.
Do you think the internet will be cut off in places? The internet will most likely be cut off at least where the war is being fought. Its probably going to be very unlikely that people will be playing videogames on SL while others are outside shooting each other, even if the internet isnt cut off. In addtion, many countries are already "enemies" with SL and have banned it from use within their borders, and I wouldnt be too surprised if more countries end up banning SL due to political tension, especially when the political tension is with the US where SL is based, and I wouldnt be too surprised if a new regime takes over and bans SL because they don't like these type of things.
Many from the other side have sims, will they be forced to abandon them? Its very well possible that they will, especially if they can no longer come on to SL to manage them. The best bet for these people would be to have a co-owner or someone take over if they forsee that something like this many happen.
How will it affect things like mp? Tax rates may change with changes in government, restrictions on items sold may change, etc.
Will it be the final straw that closes SL? Probably not. Even if SL loses a lot of people as a result its probably still going to retain enough people to keep up and running. Its US based and has many American users, who will most likely not be cut off from SL any time soon unless they choose to go to war. Most users come from western countries like the US, chances are slim people in those countries may loose SL access due to modern war, especially US and Canadian users.
Personally, I really dont think we can really tell what may happen should some war break out. With things like war, anything may happen. It can start anywhere, involve anyone, affect anything, its really hard to really tell. It all depends on the fight-where its being fought, who's fighting it, the opinions of those in charge, etc. With something like this, at least for me, theres no telling what may happen, until it actually does happen.