501(c)3 or 501(c)4? There is a difference.
The ban on political activities applies to 501(c)3 groups. It's not unusual for an organization which primarily does educational/religious/charitable activities to also have a loosely connected 501(c)4 to handle explicitly political activities.
An obvious example would be a church group with a religious and educational mission that may lead its members to disagree with current political views.
Donations to the church are and should be tax-exempt for its religious and charitable and educational work. To a **VERY** limited extent, education may include political activity — we have not yet reached the point that pastors are told they can't open up the Bible and preach against gay marriage.
However, if the church members decide they are obligated by Scripture to campaign against specific candidates for public office because of their support for gay marriage, they would be well-advised either to join an existing political organization and support it, or to form a political committee and not try to do explicit political activity with church donation dollars.
Personally I don't have a problem with saying that if a church is going to accept tax-exempt status it needs to follow the rules, and with saying that donations for political purposes don't qualify as tax-exempt. That seems to be a legitimate application of “render unto Caesar what is Caesar's.” On the other hand, it's a good idea to encourage Caesar to change his views, or in a republic like ours, to vote Caesar out of office if he harasses churches with the tax code.
If this is underlying the issues with Herman Cain, giving back the money would be a really good idea for him — and then for him to point out the well-known reality that historically black churches routinely support Democrats, and for Herman Cain to ask the question of whether any of this would have been a problem if he had been a Democratic candidate for state office, Congress, or the Senate. (In fairness, the same accusations would have been leveled against **ANYONE** running for President, regardless of party — it's the political activities of black churches on behalf of Democrats for lower offices which generally don't get investigated.)
The DLC is a 501(c)4. I am not sure about the others.