I'll have to try that, I assumed dirt roads would prevent obsidian casting just like paved roads do, but I never tested it.
Dirt roads effectively replace the shrubs, fungus, grass, plants, saplings and whatever else with furrowed soil. The furrowed soil will eventually be recolonised by organic stuff but you can otherwise cast obsidian, build anything or reestablish a dirt road over the designated area
In addition, dirt roads technically
cease to exist once construction finishes - this is why you need to periodically rebuild them, and why paved roads tend to be superior.
I'm curious, though, what type of obsidian farm you're actually using - the common
3-layer bridge model (lava at bottom, air in middle, water at top held up by bridges) produces no waste water at all and thus doesn't create any mud on which trees can start to grow.