It appears to me in the 10th demo lesson of Chess Mentor: Spassky-Petrosian, Moscow (World Championship) 1969 that 7.Re8 would lead to checkmate, but the the mentor says it is the wrong move. Does 7.Re8 not lead to checkmate? If so how?

I said this same thing  

After 7. Re8 Qd6 8. Rc8 Nc6 and black will hold on and avoid checkmate. However, 7. Re8 is also another move which would lead to an advantage for white so it has been added as an alternate correct move so that you get credited for it.

It is not an alternative move now (third of April) The mentor gives the line "7.Re8?? loses the all-important d-pawn after 7...Rfxe8 8.dxe8=Q+ Rxe8.", which of course makes no sense whatsoever.

Oh, you are talking about position 6 in the lesson. If Re8 then Rdxe8 dxe8=Q Rxe8 and black is fine. It is only after white plays Rc1-c7 in position 7 of the lesson where the Re8 move is winning.