Yes, a lease can automatically renew in Texas. Leases may include an automatic renewal clause, which will state the procedure that the tenant and landlord must go through to renew the lease (if any).
Usually, leases renewed this way will either become periodic leases on a month-to-month basis or renew for the same duration as the original lease. In some cases, the rent must stay the same (Tex. Prop. Code Ann. §§ 92.021).
The tenant or landlord can terminate a month-to-month lease with at least 30-days’ written notice, or as much notice as the lease states (Tex. Prop. Code Ann. §§ 91.001).
The information for this answer was found on our Texas Residential Lease Agreement answers.