I have a PC with a (relatively slow) burrow speed through the feat Oread Burrower. Is it possible to do any fancy combat moves from below like bull rush? What would be needed to do that, and what would be the result?
The typical creature can't attack without first having line of effect
A creature that's burrowing beneath its foe typically lacks line of effect to that surface-dwelling foe, and that means that a creature that's burrowing doesn't threaten its surface-dwelling foe (even if the creature that's burrowing has reach). (Also see Attacks of Opportunity.) Likewise, the typical foe doesn't have line of effect to the creature that's burrowing and doesn't threaten it. The surface that separates the typical creature that's burrowing from its typical above-ground foe usually completely blocks any attacks either would make against each other.
So while it would be awesome for an oread to burst forth from the earth below a foe and catapult its foe into the sky, Pathfinder makes it impossible for most creatures while they're still below the surface to make attacks against foes on the surface—even attacks like the combat maneuver bull rush. The burrower must content itself with emerging from the earth then shoving the foe the old fashioned way.1
Also, if considering the feat Oread Burrower, keep in mind that the feat grants no special sense like tremorsense to aid the oread in subterranean navigation nor in keeping tabs of its surface-dwelling foes. Without such a sense, for example, it's possible for the oread's foes to depart while the oread's undergound, and the oread won't know they're gone until he surfaces.
1 In Pathfinder's antecedent D&D 3.5e the special attack bull rush requires the attacker to enter the defender's space, therefore making a bull rush from below possible in that game. Pathfinder changed how this works, making it so that the combat maneuver bull rush is executed from the attacker's space without the attacker entering the defender's space.
I posit for your consideration Brilliant Energy. Thrusting with such a polearm (or shooting a longbow) should be perfectly doable from about 5' down (further with the bow)...
Add a ring of x-ray vision if needed.
But yeah, bull rush could be tough for the reasons pointed out by Hey I Can Chan.