QuiltE

Dampness underneath doesn't sound good and what it could mean over time.

Anything I say here has no experiential validation, as I have never used them.
My thoughts went immediately to mould/mildew and buggyboos!
Nasty Nasty stuff it could mean.

As for in-between uses, I would wonder if standing it on end, could warp the fibres and surface, and that you might inadvertently not have a smooth surface for ironing on. Rather a rippled mess. My thought would be to set it on a mesh cookie cooling rack to let it cool and dry.

But like I said ... my thoughts may be so off here.
So listen to those who have experience.

But you have given me reasons to think a bit more about this as to whether it is something I would get in the future. Or not!