Hi!
No, you don't need an application server to run a web service.
An application server may give you additional benefits, such as monitoring of deployed services, use of virtual servers etc.
It may also be possible to buy commercial support for an application server, in which case you can blame someone else when your services dies on you.
You don't even need an application server to develop a web service implemented by an EJB. There is OpenEJB which allows you to run an EJB container almost anywhere.
Best regards!