Do you think Dumbledore's Army ever had to start again? Do you think the Order of the Phoenix had to start again? Could the Order be run without Dumbledore, since he's the one that started it?

Depends if you want it to. I mean the reason J.k Rowling had not given us a lot of information about the after life of the war is so we can imagine what we want to happen to the next generation of potters and weasleys. In my imagination Voldemort will come back so yes they will have to fight. I'll have a fanfic about that on Watt pad mind you it might not be very good i'm not a professional writer:)

I'd imagine that eventually another villain would rise, and then either the Order would reform to fight him or her, if this happens within the current characters' lifetimes, or if it's later a new resistance could name itself after the Order.

Dumbledore's Army would be less likely. For this to happen, either evil would have to take over the school again, or they'd need another poor DADA teacher who refuses to actually teach.