Does bourbon improve with time after the bottle is opened?

The chemistry that defines "aging"; extraction from charred oak; only happens inside a barrel. Once bourbon is bottled, that process stops completely; there's no barrel left to interact with, whether the bottle sits sealed on a shelf for a decade or gets opened the same week.

An opened bottle isn't perfectly static, though: once the seal is broken, slow oxidation and evaporation begin, and over a long enough stretch; commonly cited as roughly a year or more once a bottle is well below full; that can shift or flatten the flavor. That's a change, not an improvement. A full, unopened bottle can sit indefinitely with essentially no change at all.