For the PSX version, this is of course already implementable in the field of graphics plugins, and is probably already doable as Aaron mentioned. For the PC version, things are a bit more complicated. Since I've toyed with the idea before though, here's my take on it.
Though FF7 does contain a lot of 3D graphics, there are at least two glaring areas which could benefit from "high-res" filtering: field backgrounds and movies. Field backgrounds are often static and thus usually will only need to be filtered once at load time, but animated backgrounds would require extra, real-time filtering as DragonZap was talking about. The movies, which are low-res and thus suffer from a pixelated appearence in high resolution, could probably also benefit from some real-time pixel filtering algorithm. This is also what I believe Gibdos was thinking about, not the 3D bits which obviously don't need filtering.
As for actually implementing it... well, that's more tricky. As I wrote in another post, we do (or at least will) have the possibility to intercept and process any FF7 graphics call and add whatever "feature" we want to it, but it will not be as easy to make any sort of high-level distinctions while doing so. In other words, we know an area perpendicular to the screen is being rendered, but we have no idea if it is a field background, a menu screen, a movie or even just a polygon facing that way. The optimal way to implement filtering is through an engine rewrite, and this is still way off.