I don't think we have to worry about the Russkis invading Canada or Alaska. What we do have to worry about is them drilling or mining or fishing or doing heavy construction in areas where we would like to do the same.

The Russians will probably establish towns, ports, refineries, and military bases in their northern littoral to support extraction and industrial activity in that area. Then they will expand into neutral or international waters--in a purely economic way, of course. But there will frigates prowling around out there, too. Eventually there will be disputes over who has a right to work in an area, or even what is allowable to do in a neutral and/or unclaimed zone, and the locations of "International Boundaries" or who has the right to pollute or seal off an area, or dump wastes, or harvest resources. And the Russkis will no doubt use lots of nuclear icebreakers and use their nukes to power industry up there. They don't care much about the environment.

This may take decades to build up to a real problem, but a big oil strike could precipitate trouble overnight.

There will be treaties and deals over who gets to do what and where, and of course, the side with the most power (not necessarily military, but certainly that too) will be able to get its way, or at least, a sweeter deal. Brute force may not always be an issue, but it will always lurk in the background.

There will be Russians, Americans, Canadians, Danes (Greenland is Danish), Icelanders and Norwegians with economic interests there, plus anyone else who thinks they can squeeze in and exploit resources in international waters. NATO will be the logical force to counter Russian interests there, but it looking pretty wimpy right now.

Not only are there enormous resources there, there are also choke points, straits, channels, strategic islands, future valuable harbors and support facility sites, all sorts of potential for conflict. And the geography changes rapidly up there with the seasons. With global warming, the coastlines are becoming more and more navigable in summer, and summer is lasting longer and longer every year. Check out that website I linked to in a previous post. The Northwest Passage everyone was so interested in during the 19th century is about to become a reality.
In another 5 or 10 years Conservatives will probably be screaming why we didn't take Global Warming more seriously.

Same as it ever was. This has been happening, in one form or another, all over the world, ever since the Bronze Age. It's why countries have navies. This is bigger than Putin, or even politics. This is a struggle for resources in a resource-scarce world.
Geopolitik.