Paul chose not to use his right to payment for gospel preaching. Sometimes he says this is in order to combat the problem of idleness (1Thessalonians 2). But in Corinth, he says it was to distance himself from other teachers who gained cudos through receiving payment (2Cor 10-13).
I can think of one example of where we might, in Australia, also choose to refuse our rights for the sake of the reputation of the gospel. One day I will give my Board of Management a headache by asking them to implement this:
Refusing lawful tax-breaks and fringe benefits for the sake of the gospel.
Many non-Christians see big churches apparently abusing the tax-breaks that religious organisations get and are scandalised. It would be very wise for churches to deliberately choose not to use our rights in this area.