Emphatically yes! How will you have proof of agreement if there is no dated signature? A signature shows that all parties have checked the inventory, agree to it’s itemised listing, have made and signed any amendments where necessary and are happy to proceed.
Once the inventory document is signed, then and only then, should the keys be handed over. You don’t want to have an adjudicator reject your unsigned inventory in the case of a dispute, so make sure it’s always signed!
