If they don't rely on volunteers to survive, if they have revenue other than donations, then you should be paid.
Ask the person emailing you/calling you if they themselves are a volunteer.
But we definitely rely on volunteers to survive, and that includes real volunteers, though most volunteers do at minimum get room and board. So yes, the non-profit picture is more complicated than the flow chart. But personally, I don't need a stipend to do what I'm doing. A warm room and good food in some of the most beautiful country on Earth is good enough for me.
Incidentally, if any Christian hacker types are interested in spending some time (3+ weeks) in the Cascades, I might have some "volunteer" work for you.
I'm skeptical of a non-profit that pays a lot of professional people for skilled work, deciding that it isn't going to pay certain types of professionals for skilled work.
For instance, my local hospital runs a volunteer League of Friends who perform regular fundraising drives for equipment such as neonatal equipment, scanners and so on. The League has an iron-clad reputation, and listing on your cv that you have done volunteer work for them actually counts as a mark of honour when applying for local jobs.