Can older military planes carry paying passengers?

The short answer is, yes, they can. The longer answer is that the FAA allows operators of older planes to charge fees for these flights through an exemption. Most older ex-military planes operate under an FAA exemption---few have normal operating certificates, instead flying under one of a few rules that allow them to be flown for exhibition purposes. When carrying paying passengers, the FAA makes organizations jump though even more involved hoops, requiring, for example, operators have documented and FAA-approved operating procedures. The Boeing B-17 that crashed in Connecticut in 2019, killing seven of the 13 aboard, was operating as such a living history experience flight. Ten of those aboard the plane were passengers. At airshows, however, rules generally prohibit operators from carrying passengers during the official air show portion of the event, and the B-17 that crashed in Dallas did, indeed, have only required crewmembers aboard.