It's obviously possible for the system to do this as you can look up the users history and see their last activity...
Personally, I'd like to see a new tag, abandoned, that is automatically applied and then used to filter questions out of the unanswered list.
Per other questions, the "unanswered" tab actually means "questions with no upvoted answer", so I don't see a conceptual problem with making it mean "questions with no upvoted answer, where the user has visited the site in the last 30 days".
When I'm looking to answer a question, I want it to know it might help someone, questions that have been abandoned are less likely to actually help someone -- even if I answer the question, it's likely that that the answer is out there somewhere for someone else to find if they are having the same problem, and that they'll find it there, and as an unaccepted answer it will appear less trust worthy if they find it on SO. As for the OP, if he hasn't visited the site in a month or more, well, he's probably either solved his own problem or moved on.
And while I can manually check to see if the user has been on the system recently, thats a lot of extra effort to put into what should be a straightforward process.