I'm a native of Venice, Florida... it's the best!
No, out of state owners are not charged extra tax on rented property.
Please contact the Florida Department of Revenue for more information on taxes, especially the transient rental tax that is collected on properties rented for less than six months. The DOR can provide you with the rules and the forms that you need.
You can also contact the Sarasota County Property Appraiser for information on Real Property Taxes.