Hello Charlie,
It is very unlikely that the mover would be damaged by leaving it engaged with the tyre, If it were to 'distorted' as per Steve W's reply, I seriously doubt the mover would be up to the job of even moving the caravan.
As for damaging the tyre, I'm not convinced it would do do any damage. A caravan tyre already carries the weight of the caravan and the profile of the tyre has to deflect to conform to the road surface. This does no damage. If you were to park on a stone, or a tree root, this does no damage, so I cannot see that the smooth form of the roller is likely to do any damage.
What may happen, is the tyre may take a temporary set, as it conforms to the changes in it circumference, These are temporary and usually after only few meters of moving a caravan the tyre walls and tread will have work softened and the memory of the deformation will be lost.
Will the application of the movers rollers reduce the attraction of the caravan to thief? I doubt it. I am sure that any self respecting thief will have already worked out how to overcome the engagement. But it might deter the opportunist.
Certainly a caravan can be towed by brute force with the mover engaged, but it will create severe wear on the tyres, and the movers rollers. Depending on the model it might also damage the movers gear boxes.
Finally you must take the advice of the movers manufacturer.