If I have a burrow speed, could I drag a creature I have grappled underground with me?
Grapple says
"Move: You can move both yourself and your target up to half your speed. At the end of your movement, you can place your target in any square adjacent to you. If you attempt to place your foe in a hazardous location, such as in a wall of fire or over a pit, the target receives a free attempt to break your grapple with a +4 bonus."
However, the rules are written from a humanoid-centric point of view. They assume you use your feet for moving and your hands for grappling. Thus even though your hands are busy maintaining the grapple, your feet are free to be used to move you and your target.
If instead you're a giant eagle who grapples with his talons and beak, your feet aren't free to move you and your target along the ground but your wings are free to move both of you into the air.
Likewise, if you're some sort of mole-man, it becomes a question of what limbs are used for burrowing and what limbs are used for grappling.
Ask your GM.