Any other colors in the non-flat areas of your image will dither on some systems, which is usually fine. Photoshop will reduce the image to 216 colors and map the image to this palette. Of course, during this color mapping some posterization may occur. Deliberately posterizing your image is a quick way to visualize your image with reduced colors and flatten your image. Using an adaptive palette with diffusion dithering generally gives the best results.
Perceptive palettes are even better.
However, on 8-bit machines browsers will remap the adaptive palette into the client's system palette. Usually the results are acceptable. While the entire Mac system palette will be non-dithered on a Mac, Windows users will find some colors dithered. Use Undo or Revert to tweak until you're satisfied with the image. Remember, your graphics will be viewed on screen at 72 dpi for an instant, not printed on a reverse dye-sublimation printer so you can get away with less bits than you may think.