Quick answer: zero Texas state tax — the state has no income tax at all. Only federal capital gains tax applies, and the stepped-up basis usually keeps that base small too. See the full cost picture in the Texas probate real estate sale calculator.
No state income tax, period
Only the federal side to plan for
Federal capital gains tax still applies, but under IRC § 1014, an inherited house's basis resets to its fair market value on the date of death — sell soon after death, near that same value, and the taxable gain is small or zero even at the federal level.
Selling costs reduce the federal gain
Agent commissions, closing costs, and property preparation costs incurred in connection with the sale reduce the capital gain (or increase the capital loss) on the federal return — worth tracking carefully even though Texas itself has no equivalent state deduction to claim.
Watch the homestead exemption's expiration
If the house carried a homestead exemption, it expires at the end of the year of death, or whenever the property stops being used as a homestead. The new property tax bill without that exemption can be significantly higher — a real cost during the estate period, even though it isn't a capital gains issue.
No estate or inheritance tax either
Texas has no state estate tax, and its state inheritance tax was repealed effective September 2015. Only the federal estate tax applies, and only to very large estates.
A local probate attorney can review your estate — many offer a free consultation.
The absence of state income tax applies identically whether the sale closes in Harris, Dallas, Travis, or any other Texas county — there's no local income tax layered on top anywhere in the state.