Trump is encouraging pastors to push politics. Here's how that's gotten them in trouble before.

Up for Senate reelection more than 60 years ago, Lyndon B. Johnson became gripped by fear that nonprofits were pumping large sums of cash into efforts to elect his opponent, a millionaire rancher and oilman named Dudley Dougherty.

So Johnson, a Texas Democrat, got creative.

In 1954, he pushed a...