The Farmers' Co-op in Sheridan moved our 1000 gal tank a little ways (to re-position it between the new house and the shop) when it had about 600gal in it. They have a propane tank trailer, which is a frame that allows them to put a couple of slings (not chains) under the tank about 3' in from each end and hoist it up on electric hoists. The hoists were at or near capacity and the 3" square steel tube frame wasn't real happy lifting that amount of weight.

I would not try to grab a full 1000' gal tank by the lifting hooks on top of the tank to pick it up.

Propane goes about 4.11 lbs/gal at 60 degrees F. Considering that a 1000 gal tank is rarely filled beyond 80% capacity (to allow expansion room in the heat of summer), so you're looking at something like 800 gal * 4.11 = 3,288#, plus the weight of the empty tank, which for a typical ASME steel tank is about  1800#.