
The Artificer |

I would like to know if my character can stay mounted while their creature companion is burrowing.
Absolutely, check out this rule from the Survival skill.
"You can use Survival to ride an animal or another beast as a mount. The most typical actions while riding a creature don’t require checks. As a move action, you can either mount, RIDE (using the mount’s speed instead of yours), or dismount from a creature; you must use one hand to ride a creature."

Metaphysician |
Technically, that rule doesn't say anything about being able to use any and all movement modes a mount has freely. It just doesn't forbid it, either. Something that isn't forbidden isn't automatically possible, just like something that isn't explicitly allowed isn't automatically forbidden.
I'd rule on a case by case basis, but generally speaking, if a creature actually does have mount rules for it, there probably should be *some* way to ride along while it uses Burrow movement. Maybe it requires special tack to do so, or maybe such is optional but desirable because going without means making more/harder Survival checks to avoid taking damage from the ride.

Wesrolter |

Lie in Wait (Ex) While mounted on your eshar companion, you can remain burrowed in sand or loose dirt for up to 1 hour, and if combat begins during this time, you gain a +4 circumstance bonus to any skill check (normally Perception or Stealth) to determine whether you act in a surprise round
The Eshar creature. If you couldn't use the burrow ability then this would be pointless, so I would go against my own initial view to say yes, you can use burrow