No. Resolution DOES affect performance. The higher the resolution, the more data the video card has to process. Color depth matters too.

The number of physical pixels on any given monitor (LCD anyway) is constant, but the number of pixels that are used to represent the computer interface is very much variable.

This is really pretty easy to see. Take any random game that doesn't use a fixed resolution. Run it at say 640x480 and measure FPS, then run it at whatever the highest resolution is your monitor supports and measure FPS again. You should see a noticeable dropoff.