The visible portion of the NTSC NES or Super NES picture is 256x224 with 8:7 pixel aspect ratio, or 256/224*8/7 = 1.31:1 display aspect ratio, close to the 4:3 of classic CRT SDTVs. I can think of two ways to get closer to the desired 1.77:1 DAR of HDTVs without compromising proportion: expand the visible area to 336x224 or crop to 256x164.
Zooming in
PocketNES, an NES emulator for Game Boy Advance, had to squeeze the screen into 240x160 pixels. It had a
scaled mode that would compress 213 lines into 160, looking much like what you get when you stretch the standard NES picture to fill an HDTV. It also had an unscaled mode that would follow a location in the NES's RAM that stores the vertical coordinate of the player character and pan the entire emulated display up and down based on this position.
Ports of NES games to Game Boy use essentially the same technique, often with a rearrangement of the status bar to open more space for the action. These include Balloon Kid (GB port of Hello Kitty World for Famicom with all the Sanrio trademarks filed off) and Super Mario Bros. Deluxe. And in some cases, the status bar is made transparent and overlapping part of the playfield, as in Super Mario Advance 4 (GBA port of Super Mario Advance 3).
Zooming out
The NES background display can be set to 512x240 pixels ("horizontal arrangement" or "vertical mirroring") or 256x480 pixels ("vertical arrangement" or "horizontal mirroring"). Scrolling operates by
updating the part of the background plane that's offscreen with tiles that will soon become visible.
Animation of updating in Super Mario Bros.
Games using vertical mirroring, such as Super Mario Bros. or Contra, may allow a cheap modification to support widescreen operation. Running the PPU at four pixels per CPU cycle instead of three would allow drawing an extra 40 pixels of background on each side totaling 336x224. Games that make their updates near the edge of the screen, rather than in the center of the offscreen part of the plane, will need to be modified. And as rainwarrior mentioned, sprites will pop in and out of visibility as they go through the extended background.
There is one slight inaccuracy in the animation above: Super Mario Bros. is actually well behaved in this respect, updating in the middle of the offscreen area rather than near the edge.