They should do their due diligence in identifying current or potential moisture issues and advise on re-occupancy timing for the tenant. 

The moisture around the toilet made the glue unstick and peel up.

This will force away ground moisture to pump bucket and out of the house.How wet is it?

On top of that I would do a termicide soak application, then a layer of moisture barrier. 

You will get burned out motors.The tenant will have higher utility costs.That air flow and air quality will be affected.The tenants could claim the unit was not working properly making you eat the cost of a service call.They could also complain of moisture problems because the property wasn't breathing properly with air circulation.Filters are cheap and repairs are expensive.I definitely would not leave it up to the tenants.

You could also have moisture ants or other.

I originally wanted the same in the kitchen, but was warned against it at the time due to possibly moisture issues with tenants doing this or that in the kitchen.

They will prevent more costly repairs from moisture damage.  

If you put a screen on the pipe, and it's cold out, moisture in the air can start to freeze when it hits the screen, which eventually will block the flow of air. 

She has never mentioned to me that the moisture came to her daughters room.