1 Insurance to cover the piano move should be high enough to cover its full replacement. 2  Costs vary by piano moving company and run $7-$10 per $1,000 of value. 3  When a piano is moved as part of an entire household, the insurance rate may be about $3.50-$4 per pound. Climate controlled storage runs $50-$60 per month. 2  A piano that needs to be lifted through a window by a crane can cost $750-$2,000, depending on access. 3  A 21x62 window opening is needed for a grand piano. 4  Insurance to cover the piano move should be high enough to cover its full replacement.