The general answer here is... it probably won't change much. The ISS has been around since 1998. Since then Russia has invaded the Republic of Georgia (2008) and seized Crimea from Ukraine (2014). NASA put out this statement in regards to the 2022 invasion of Ukraine
"NASA continues working with all our international partners, including the State Space Corporation Roscosmos, for the ongoing safe operations of the International Space Station. The new export control measures will continue to allow U.S.-Russia civil space cooperation. No changes are planned to the agency’s support for ongoing in orbit and ground station operations," NASA said in a statement today that agency spokesperson Joshua Finch emailed to Space.com.
Will we see a situation like the one in the film 2010 where the Soviet and American crews were ordered not to interact with each other?
There's a key difference here. In the movie, you have a separate Soviet spacecraft and an American spacecraft. Neither had to interact with the other to function (fun fact: it was a Soviet mission to the American spacecraft that got them there in the first place).
The ISS has both American and Russian components and you need both parts to make the ISS work
"The Russian segment can't function without the electricity on the American side, and the American side can't function without the propulsion systems that are on the Russian side," former NASA astronaut Garrett Reisman told CNN. "So you can't do an amicable divorce. You can't do a conscious uncoupling."
All of this relies on there not being a "hot" war (i.e. both sides are fighting with one another). If that were to happen, I would say the worst-case scenario would be one side abandoning its part of the ISS. That wouldn't immediately threaten the ISS (both sides can send personnel and supplies up by themselves), but the lack of ongoing support for the other end could hamper its operation.