The answer is technically no there is no requirement that the browser reside on the server (and lots don't ever access the client from the server so have no need for a browser), however I have found it quite usual and beneficial to have a browser on the server for the purpose of quick actions being performed on the server (such as site admin actions when logged into the server) and also for trouble shooting purposes when connectivity issues occur and there is a question regarding the browser blocking internet access through a firewall, anti-virus, etc action and not hosting the service to the network. The correct workings of the server can be verified by using the localhost as url and this alleviates the hosted URL redirects and other things that may be occurring. One can verify and see that the services are running correctly and the application accessed directly on the server and proving that if issues are occurring then it is a hosting to the network issue and not an application issue on the server. So although it is not "required" to be on the server, the browser can in this instance serve a very good trouble shooting tool.
Hope this helps,