A lot of the buzz around Lisp web applications does relate to mutable state accomplished through continuations, but that's not a requisite to using Lisp, nor is it the only advantage to using Lisp.

I manage state via relational databases, and haven't had to deal with any memory-leak bugs. I even found it easy to prevent leaking of database connections.

Mutation is definitely optional. A lot of people even still count Lisp dialects among the functional languages.