No, not at all. Static websites with no dynamic data will not require any data connection. Or, for some web applications data can be stored directly as a static file in a folder system (such as XML or even directly as HTML).
However, for any project where multiple users can log in and change content regularly, having a database will make it far easier to scale.
Using a web server and a data server together through applications is the standard way applications run, and finding the “right” combination of technologies for a project is a process that takes patience and a joy of learning.