Can the tour play through the rooms by itself, on a screen?

No: Panoramr has no autoplay tour that walks from room to room, and no kiosk mode. What a screen can do is turn each room's view by itself: switch on Autorotate on idle on the Design tab and set its speed. The view stops when someone drags or zooms the picture and starts again 3 seconds later. The tour opens in its first room: How do I change the order of the rooms? Auto-rotate a tour on a kiosk or display has the details.