NATO is an organization for the mutual protection of Europe. Created after the Second World War its mission was at first to have an organization to contain the Soviet Union. Currently my belief is that there mission is to offset Russia's desire for expansion. If NATO were to involve itself in the Middle East it could only do so to protect a constituent member, in this case Turkey. Russia might see this as an attack against her interests and anything could happen after that, including war with Russia. While Russia's Army isn't what it once was, and their Navy is a bunch of rust buckets. They certainly have tactical nukes. Complex bad idea.