Like the song says, there's a time for everything - and this includes the choice to make a database-driven website or just a static one. There's too many times that I've seen people create a dynamic website when all they really needed was a few static pages and not much else.
This article from Builder.com talks about just that. There's times when a dynamic website is just fine, but a database doesn't need to even get involved with it. All they need is maybe a few things from a text file and some updated stats pulled from an external site and they're all set. This article, though shorter, gets right to the point when asking "When do I need to use a database?" Ultimately, of course, it's up to the designer and coder, but there are instance that clearly don't call for a database. You just have to get used to what's what and knowing when to stop those over zealous proponents who think that datbases fix everything.