My understanding is as follows... (Tested with BIG-IP v14.0) If a monitor marks all pool members in a pool down (offline for traffic processing - red diamond), then the pool is also marked down, as is any associated virtual server that uses that pool exclusively (meaning it has no other pool it can use for load balancing).
If the virtual server has more than just the one downed pool available for load balancing, perhaps via an iRule or a Local Traffic Policy (or a combination of default pool and an iRule or Local Traffic Policy) then the virtual server will not be marked down. However, any persistent or new connections to the virtual server that wind up being directed to the downed pool (either via the iRule, Local Traffic Policy, or default pool configuration) will fail.
No new or persisting connections are allowed to a virtual server that is marked down (red diamond). Existing connections to the virtual server and downed pool members may or may not complete normally, depending on what the underlying problem is that caused the monitor to mark the pool members down to begin with.