<p>What are the downsides to using static methods in a web site business layer versus instantiating a class and then calling a method on the class?  What are the performance hits either way?</p>

The performance differences will be negligible.
The downside of using a static method is that it becomes less testable. When dependencies are expressed in static method calls, you can't replace those dependencies with mocks/stubs. If all dependencies are expressed as interfaces, where the implementation is passed into the component, then you can use a mock/stub version of the component for unit tests, and then the real implementation (possibly hooked up with an IoC container) for the real deployment.