Suppose a civilian buys a military jet, for example an T-38. Would he be allowed to use afterburners?
No, no restrictions on using an afterburner. There are a few F104 Starfighters on the civil registration that must use the afterburner to take off.
However, most countries forbid supersonic flight by non military operated aircraft in their airspace. The only time Concorde went supersonic was over the ocean.