COLORADO SPRINGS, Colo.—The conservative donor network founded by the billionaire Koch brothers won’t support a GOP contender in a close congressional midterm race, saying he doesn’t follow Koch ideology on spending and tariffs.

Network leaders, who promote small government and are against tariffs, said Monday the group will sit on the sidelines in North Dakota, where the prospect of a trade war is worrying export-dependent farmers. Democratic Sen. Heidi Heitkamp is vying for a second term in a race against Republican Rep....