Depends on the game. 3D games are resolution-independent which means the game has a bunch of 3D objects with levels of detail (polygons) that stays the same regardless of screen resolution. In these cases all screen resolution does is determine the number of pixels used to draw the 3D objects on screen. 2D games, and especially 2D games with pixel art like you see in most 70's to 90's era games, rely more on specific screen resolutions to draw properly but like 3D, 2D game objects also have individual levels of details(pixel resolution) that can be scaled up or down to fit the target screen resolution.
Your gfx card has a selection of screen resolutions that it supports, usually in the range of 640x480 to 1900x1200 or something for newer cards. When you set a game to run in one of these resolutions you're telling it how many pixels to use to represent the graphics on screen. But all the rendering calculations, physics etc take an equal amount of time to process. So in terms of strain on the CPU and GPU game objects have little to do with the screen resolution itself.
Hope that helps?