I use a cover from ADCO that has a "breathable" roof section, but, only when the TT is laid up over the winter season. I use the same type of material on the covers for my two antique cars (well, one's an antique the other is currently a "classic). No moisture problems to date and in the winter time, it keeps puddled water from freezing in and under the edges of the sealants used on the lap joints (front edge) and along the air conditioner's foam gasket. The thawing and freezing, expansion and contraction of water has the same affect on sealants as it does on roadway surfaces...living in areas where it doesn't ever freeze would, of course, eliminate the problem, but, constant sun (UV) exposure can be just as damaging to the sealants and plastics found on many modern TT roofs...just like your tires, if you aren't going to be using it for a long period of time, IMHO, a cover can only help in making the roof components last longer and keeps the TT cleaner, minimizing the aging affects of airborne pollutants.
My 2 cents, your mileage may vary...
2 Cats; J-Lo and Ragamuffin :R
2014 Thor Tuscany 40RX DP
2011 Ram 2500 Longhorn CTD HO
2011 Keystone Cougar 318SAB (now gone)
2008 FunFinder X 210WBS (Sadly gone)