It seems they have to pay if they have income from non-political activities. In this example, renting space to an unrelated entity.

The political organization taxable income equals its gross income (excluding exempt function income) less deductions allowed by the Code that are directly connected with producing gross income (excluding exempt function income), computed with certain modifications set forth in § 527, including a specific deduction of $100. The exempt function income of a political organization is income that the organization sets aside for use for its exempt function that was received as one of the following four types of income: (1) a contribution of money or other property; (2) membership dues, fees, or assessments from a member of the political organization; (3) proceeds from a political fundraising or entertainment event or from the sale of political campaign materials, which are not received in the ordinary course of any trade or business; or (4) proceeds from conducting bingo games that are defined in § 513(f)(2). Thus, investment income, or income from a trade or business (such as renting excess office space to an unrelated organization), of a political organization is not exempt function income and is subject to tax. Taxable income includes exempt function income (such as contributions) for any period of time that a political organization does not file a Form 8871 as required.

This is from here: https://www.irs.gov/charities-non-profits/political-organizations/taxable-income-political-organizations

I am not a tax accountant or tax lawyer.