As a web programmer with a degree in Fine Arts... YES. They do.

When you really get down to it, the web is about using visual media to communicate ideas. So is Art. Likewise, both disciplines have a heavy technical component. Both have mixed audiences... most people care about the tech side, some just want to experience the final product. Both disciplines also start with a blank page, and build it into the final product. The final products of both disciplines are greatly improved when the audience is engaged, and can be revolutionary when the audience collaborates.

So do web designers need to understand art? Yes!