"Do I need to file Form DR-405 Tangible Personal Property Tax?"

That depends upon whether you have tangible personal property, as defined by the property appraiser for the county in which your rental is located, that is used in connection with your rental. If you rent an completely unfurnished unit (condo or house), then there should be little, if any, tangible personal property to report. 

Regardless, filing a Form DR-405 ensures an automatic exemption of the first $25,000 in assessed value of tangible personal property.

Form DR-405, if one needs to be filed, does not have to be filed via TurboTax. In fact, at least one county requires the form to be filed online from the county's web site and several others have their own, modified, version of the form. It would therefore be wise to consult your county property appraiser's web site before proceeding. 

A list of county property appraisers' sites can be found here: http://floridarevenue.com/dor/property/appraisers.html