Is it a bad idea to use a bitcoin wallet address in a domain name or URL? Aside from it forever being public and kind of marrying a business to a wallet.
If you mean as part of a URL or domain name, then yes, I think it's a bad idea. If your private key is somehow compromised (eg through a stolen or hacked PC) you may no longer want to use that address, but your users may have it in their bookmarks for years afterwards. 

However, as long as it didn't have any particular meaning (such as being accompanied by an instruction to donate to that address), it may not cause any harm. Secondly, it's easy to redirect a user's browser to a new page if you need to.

Usually URLs are best kept short unless it's a sub-section of a site used for sharing or bookmarking. Keeping them looking like words (as much as possible) helps with usability, so unless your vanity address is the longest ever generated and you want to show off, I can't see a good reason to include it there.