WASHINGTON—The tug-of-war within the Republican Party that helped end Rep. John Boehner’s career is likely to intensify this year both on Capitol Hill and in the tumultuous GOP presidential race.

The House speaker’s announcement Friday that he would leave Congress on Oct. 30 isn’t expected to mollify either the House’s most conservative faction, which is determined to take an unyielding stance in the face of fiscal deadlines, or dissatisfied GOP primary voters rooting for outsiders who have pledged to uproot Washington politics....