No, that is not true. There will still be property taxes to pay but at 65 you become eligible for additional tax exemptions.
Texas law requires school districts to offer an additional $10,000 residence homestead exemption to persons age 65 & older or disabled. A city, county, school district or special district, can adopt a separate residence homestead exemption for persons age 65 & older or disabled, this will differ for each entity.
The Tax Code also places a ceiling on school taxes for residence homesteads owned by persons who are age 65 & older or disabled. School districts and entities other than schools who adopt a tax limitation may not increase the total annual amount of ad valorem tax it imposes on the residence homestead of an individual 65 & older or disabled, above the amount of the tax it imposed in the first tax year in which the individual qualified that residence homestead for the applicable exemption. Please note that tax limitations do not apply to land, buildings, structures &/or fixtures not included in your homesite.
Homeowners age 65 & older or disabled who may also transfer the percent of tax paid, based on their ceiling, when they purchase another home and use it as their principal residence.
Tax Ceilings do not apply to Non-Homesite properties.