An insurance company will expect you to have been searching for a vehicle to replace yours if they feel it's a total loss. The moment the insurance company has made an offer to you on your vehicle, they will likely stop paying storage fees. This will happen even if you do not agree with the offer. You could be left with a rather large bill if you ignore their attempts to settle with you.
If the insurance company is still investigating fault, they will probably ask for your permission to move the vehicle to a facility that does not charge daily fees for the storage of your vehicle. There is no reason not to allow them to do this. If you refuse their request, you will likely be racking up unwanted storage fees.