In Season 3 of the Sopranos. Paulie and Chris let a Russian associate escape in the woods in the episode "Pine Barrens".
Was there ever any reference to the character or what happened to him in any later episodes? I can't remember there being any.
No, he was never referred to again. I assumed that he got away and kept his head down from there on out.
In an interview with Terrence Winter and David Chase it was revealed that they kept it intentionally vague (emphasis added):
Perhaps the biggest legacy of ''Pine Barrens'' is what you don't see, i.e., what becomes of the Russian. It has become one of the most tantalizing mysteries in The Sopranos history.
WINTER: That's the question I get asked more than any other. It drives people crazy: ''Where's the Russian? What happened to the Russian?'' We could say, ''Well, he got out and there's a big mob war with the Russians,'' or ''He crawled off and died.'' But we wanted to keep it ambiguous. You know, not everything gets answered in life.
CHASE: They shot a guy. Who knows where he went? Who cares about some Russian? This is what Hollywood has done to America. Do you have to have closure on every little thing? Isn't there any mystery in the world? It's a murky world out there. It's a murky life these guys lead. And by the way, I do know where the Russian is. But I'll never say because so many people got so pissy about it.
From the 12/5/2012 Slate article David Chase Doesn’t Care About the Russian:
“Who gives a shit about this Russian?” David Chase says. The creator of The Sopranos has never understood his audience’s fascination with Valery, the Russian mobster who disappeared in the legendary “Pine Barrens” episode. It was a one-off story that needed no closure, Chase says now. He recalls thinking, “We did that show! I don’t know where he is! Now we’ve got to go and figure that out?!?!”
Terence Winter, who wrote “Pine Barrens” and many of the series’ other memorable outings, agreed with the fans on this one, much to Chase’s frustration, and kept pushing his boss to add a coda to that story in The Sopranos’ final season. They finally hit on an idea everyone would be happy with: Tony and Christopher pay a visit to the local Russian mob boss, where they find Valery sweeping the floor, not recognizing Christopher thanks to a traumatic brain injury suffered when Chris and Paulie were shooting at him. (It would be explained that a local Boy Scout troop found him with part of his skull missing, and saved his life.) At the last minute, Chase changed his mind, and he recalls a despondent Winter insisting, “God, you’re making a huge mistake leaving that on the table!”
It's non-canonical, sure, but we'll take what we can get.
"Was there ever any [later] reference to the character?" Yes, in a later episode we see Chris and Pauli in one of the backrooms, retelling the highlights of the story to the other "Made Men" - during which time Chris openly alludes to it having been all Pauli's fault, because he "lost his cool" and broke the remote-control. Pauli is aware of how dangerous that could be to his reputation, and therefore denies it. But no: We, the viewers, never receive any additional information (above and beyond what we ourselves saw in "Pine Barrens").