So far my stove has been shut down for 3 months, maybe even 4 already. I tore it apart and vacuumed it very well within 24 hours of shutting it off. Then all I did was close up the openings fairly well and place a "Damp-Rid" bucket it the bottom of the stove. I looked inside yesterday and the Damp-Rid had about 1/2" of water in the little catch cup, and the inside of the stove looks just like the day I cleaned it - maybe a few specks of surface rust here or there.
The stove is in the basement, which is quite dry, and fully finished. There is a dehumidifier down there and also it's tapped into the HVAC system, so it keeps it dry with the A/C. Even still, there's enough moisture getting into the stove to activate the Damp-Rid. Good stuff!