I'd like to jump in and (hopefully) add to this conversation, regarding the OP's idea.
If you could decode the tile maps and the placement of items/enemies in the level, it's possible you could do what you're thinking. However, depending on if the background has any kind of effects going on, enemy AI, and other things, you may or may not get what you're picturing in your head. For example, some enemies are not activated till they're within a certain distance from the screen's draw area. Even if they were activated, they might just sit there till you're within the activation distance.
Graphically, something like Zelda might work, where there's nothing special going on with the tiles on the backgrounds other than maybe some tile swapping or palette swapping for animation, however on something like the Sonic games, the background relies on hblank scrolling (did I get that right? it's been a while since I worked in that industry), which may render 'ok' on the horizontal pixels around the original's resolution, but might look really bizarre above and below that. It's even possible that you would get a background image that's not big enough for the target resolution because it was never intended to go beyond the native resolution.
Were I to guess, this might need to be done on a per-game level. Depending on the software and the game engine used to make the game, one company may compress their data differently from another, and even the same company may use a different system for the very next game.
I imagine this kind of thing is not impossible, but it might not be worth the effort in many cases. Like the others have said, you may see something you weren't supposed to see. You might see secret areas you never saw because you didn't know they were there (maybe good) or you may see broken bits of levels where the level designer left parts to clone to other areas (maybe bad). Maybe you might see a message a designer left for hackers (something I'd thought about doing at one point for games I've worked on).
Anyway, I'm rambling now. Bottom line, not impossible, but perhaps too much trouble for too little gain. It might be a fun way to play a tired old game though.