Yes, there’s a good reason that websites don’t use Courier often in websites. Sans-serif fonts, such as Arial and Helvetica, are more easily read on computer screens, while serif fonts (like Courier New and Times New Roman) are better for print on paper. I don’t know exactly why this is, but it’s true. Especially considering Courier New is a monospace font (typeface in which each letter takes up the same amount of horizontal space). that makes the kerning such that it’s even harder to read on a computer screen unless very large.
However, in a sense, almost all web design uses Courier in a way. Monospaced fonts are preferred for coding, such as HTML and Flash’s ActionScript. In fact, if you are using FireFox and go to View -> Page Source, the HTML/CSS code should be in Courier.
Hope this clears things up a bit. :) I guess my typography classes haven’t been completely useless after all.