A common tactic in old mines was to set up multiple pairs of shafts. One shaft would have a fire lit at the bottom of it. The hot air in that column would rise, forcing cold air to flow down the paired cold shaft and setting up a flow of air through the mine.
How does this help you?
Well, any way to set up a temperature difference can create this kind of airflow. Sunlight on one half of a mountain (or mirrors to direct sunlight to the base of a shaft if no mountain is suitable), a subterranean river that’s warmer than the air, geothermal activity if your dungeon is big enough, even the body heat of your creatures if they’re clustered in one place; all of these could set up permanent airflow with no moving parts needed!
If the dungeon is designed and set in a location with a prevailing wind direction (say somewhere like a valley in a steep mountain range) then pairs of entrances can be built with open doors facing into/out of the wind to achieve the same effect. The wind blows into one entrance, circulates, then blows out the other. If you don’t want these to be ‘entrances’ per se then you can make small buildings with one wall missing atop a ventilation shaft,
One thing to watch out for with this kind of ventilation is that the setup of the dungeon matters. Opening/closing the wrong sets of doors might lead to chunks of dungeon that are cut off from ventilation, or might just stop it working altogether! The basic requirement is to keep the dungeon split into sections with an ‘in’ at one side and an ‘out’ at the other and to minimise airflow between the sections.
This gives your dungeon a reason to have a bunch of seemingly randomly placed doors, some of which are open, some closed, and some locked.
It also means a rampaging band of adventurers is likely to wreck the ventilation, so watch out for that!