No. Neither the Texas Property Code nor the Texas REALTORS® Residential Lease would require a landlord to replace or repair something like this.
While Paragraph 18D(1) of the TAR Residential Lease states that the “landlord will pay to repair or remedy conditions in the property in need of repair if the tenant complies with the procedures for requesting repairs,” this does not mean that the landlord has the obligation to make every requested repair. Paragraph 18D(2) of the TAR Residential Lease states that a landlord will not pay to repair “items that are cosmetic in nature with no impact on the functionality or use of the item,” and a landlord could argue that worn carpet falls under this category. Additionally, Paragraph 18C(1) the TAR Residential Lease states that all decisions regarding repair will be at the landlord’s sole discretion.