I'm going to design a DMVPN full-mesh network same as following topology.
If hub-router which has static public IP address shut down, would another three routers be able to keep their tunnels?
The answer is no, as the nhrp mappings will change and there will have to be a change in the hub. You could configure a dual hub scenario where if one goes down the other can take over.
assuming you have a pretty decent code
i do not think anything would happen, the tunnel will continue to be up
but just make sure that the duration of downtime is less than the time remaining for the tunnel to reestablish
depends on whic hphase of dmvpn u r running
what code r u running on your routers