So, my question is if Putin or Russia get anything out of this. Is there anything they get out of it? Sales of weapons to Iran? Isn't Iran allied to Russia?
Russia might gain in that because of sanctions on Iran, Russia is a major destination for Iranian oil. So, the current situation is that the Iranians are happy that they call sell their oil to Russia at below market rates. So the US-Iran spat benefits Russia economically. If US/Iran relations thaw then Russia loses their special trading status with Iran.
EDIT: realpolitik is also at play. We shouldn't believe for a second someone wouldn't do something to an ally that starts a war between that ally and their enemy. Being on top often means destabilizing other relationships so that yours is the primary relationship. Gotta keep those 'allies' in line somehow. When they start working between themselves - that's a threat. That's also why Trump and Co hate NATO and the EU and the United Nations so much, and also why they oppose
regional free trade agreements. They're all networks for countries to sort things out without needing the USA. The nationalist/populists want their to only be one type of agreement - unilateral singular agreements between the USA and individual other countries, and they want to undermine all other agreements between other countries, allied or not.