Can you wear a red shoe with a black and white dress?

This depends greatly on the style of the dress and the pattern. If the dress is equal parts white and black, and also ends above the calves, a red, chunky heel would look very nice and add a little bit of flair to a sophisticated outfit.
However, a long black and white dress with red heels would look awkward. If the dress is long I recommend wearing black heels with it.

If the dress is also considered a "statement" dress (a bold, eye catching pattern) then it may be best to leave that as the center of attention and not clutter up your outfit with red shoes.

I've included great examples of black and white dresses with red heels in the sources and related links tab below.