When a pool member is marked down bigip will not open any new connections to it. Also, by default, bigip does not close any open connections to a down pool member (this behavior can be changed with the pool setting "action on service down").
When all pool members are down (red diamond status) then the pool and virtual server will also be marked down. A down virtual server will still accept connections and requests. This allows bigip to send a redirect (for http) or respond to client in some way (for example using an irule to provide an http response "Site is temporarily unavailable"). The down virtual server will process requests but they are not sent to any down pool members.