You wouldn't want to apply it to the entire screen, but rather to individual sprites and background images separately. The design of the graphics hardware on the NES and SNES should make that mostly possible for the majority of games.
Unfortunately, applying an effect separately to sprites versus background layers isn't as useful as it sounds; a lot of games use non-animated sprites to touch up the grid-like nature of background tiles (so treating them separately would look weird), or use multiple sprites to make up a larger character (so treating sprites individually would look weird). I think one of the NES MegaMan games even has a boss battle where the boss is actually a background, and the platforms MegaMan stands on are sprites.
You wouldn't want to apply it to the entire screen, but rather to individual sprites and background images separately. The design of the graphics hardware on the NES and SNES should make that mostly possible for the majority of games.