WASHINGTON—The House adopted a budget Thursday that sets the stage for a rewrite of the tax system, as Republicans overcame internal divisions and began a six-day countdown for releasing closely guarded details of their tax plan.

The vote on the budget for the fiscal year that began Oct. 1 was 216-212, a slim margin that is an early indication of the difficulty Republican leaders will face in cobbling together the votes for a tax bill packed with trade-offs.

Eleven...