Although it varies depending on the shelter, most rescue groups require adopters to provide photo identification and a copy of a lease if they are renting. Shelters often provide information their websites regarding what paperwork they require.
Some shelters require proof of current address as well. This can be a utility bill, bank statement or other mail with an address on it. Some may request additional documentation from renters to prove that their landlords have given permission for the new pet. This may take the form of a letter giving permission, or the shelter may ask for the landlord's contact information to call directly.