In all the years of dealing with travel trailers, ever since I was an 8 year old kid and my parents got a TT, never removed the propane tanks.
On the other hand, my mother had a 20 pound propane tank in the basement of her house off an old BBQ grill. It was half full/empty. I wanted another tank to use for a Coleman gas stove. She gave it to me. After all those years, it worked perfectly fine.
When it finally ran out of propane, I ended up trading it at a swap-em out gas station and that's how I got the newer one with the triangle knob, instead of the old style rounded knob.
Actually, to store them inside, or keep them on the camper exposed to the elements over the winter months, I don't think it matters one way or the other. It's really all about your own comfort level and potential theft than anything else. But for the life of the tank, I don't think it matters.