In one place I'm familiar with, Corvallis Oregon, a free Outdoor Burning Permit is required through the Corvallis Rural Fire District. The very top part of the applications states:
Designation of burn days is based upon daily environmental conditions. Between October 1 and June 15, the Oregon Department of Forestry determines whether burning will be recommended. Between June 16 and September 30, the Oregon Department of Agriculture makes that determination.
The Department of Forestry is more concerned about forest fire conditions during the normally wet season, the Department of Agriculture is more concerned about air quality (yes, particularly in the Willamette Valley burning of fields used to fill the valley with smoke in the late summer).
Besides stating a day (and time frame) and the property address, the rest of the form (link at Corvallis city site, one needs to read the handout from the State of Oregon Department of Environmental Quality before getting to the permit itself) is a bunch of checkboxes acknowledging all the requirements one has to follow. This includes checking the day of your permit if burning is allowed in the area that particular day.
It is free, so revenue is not the issue. The fire department proper does not handle determination of allowed burn days, that is handled by the state. Whether a burn day is allowed or not is a combination of forest fire danger and air quality considerations. The purpose of the permit seems more to be to make sure that the landowner understands (or at least checks the boxes saying they understand) all the requirements to safely burn, and all the various fines they may accumulate if they don't do the right things.