Do you keep paying once you buy minecraft?

No you buy it once with all the money and play with no extra charge.

No you don't pay anything after that

Nope once off payment!
However some greedy server owners tried to make it pay to win such as selling perks,
But notch said HELL NO, and made it illegial to make servers pay to win.

No, why would you need to keep paying

No Minecraft is pay once. Don't lose your account information though (such as giving a fake email) as you will need your account information to play.

Note that some private servers want their players to donate to them to keep the servers going. These are not official Minecraft purchases and they have no consumer protection.