What is the weakest file on the board? If f2 and f7 are the weakest squares on the board, if the f-file the weakest file on the board? For example, other than attacking f2 and f7, is there a theoretical advantage in dominating the f-file?
The f2/f7 squares lose some significance after castling has occurred. The square is no longer protected by just the king - the rook now also protects it. But a f2/f7 pawn with a king on g1/g8 might be pinned by a bishop, which sometimes results in tactical opportunities.

Generally, you should attempt to control the center of the board. The f-file is not the center, but it is adjacent to the center. It might make sense to try to control the f-file if it is open or half-open (or if you plan to make it that way) but that's true of any open file. 

If the file is not open or half-open, then it probably makes more sense to look at the individual squares rather than the file as a whole. If White has a pawn on e4, that pawn controls f5, but it also means that pawn can never control f4 (so f4 might be weak.)