Given the Trox multiple upper limbs and and burrow trait, does he need to sheathe/put down his weapon before burrowing, or can he burrow all the same?
Pathfinder makes no reference to hands needed to Burrow. Even creatures other than a Trox can typically Burrow (if they have the ability) while armed.
Barring GM's making their own decision on the matter, we have very limited rules relating to Burrowing.
First, from d20SRD.org (because Pathfinder did not copy the OGL passage, nor make their own):
A creature with a burrow speed can tunnel through dirt, but not through rock unless the descriptive text says otherwise. Creatures cannot charge or run while burrowing. Most burrowing creatures do not leave behind tunnels other creatures can use (either because the material they tunnel through fills in behind them or because they do not actually dislocate any material when burrowing); see the individual creature descriptions for details.
Or, the Trox's own Burrow trait:
Trox have the burrow movement ability. (3 RP) They gain a burrow speed of 20 feet.
Which is pretty straightforward.
However; the best evidence is probably the Burrow spell. It is Paizo material and contains the following notes:
Using burrow requires only as much concentration as walking, so the subject can attack or cast spells normally. The burrowing creature cannot charge or run.
This seems to imply that you have no problem holding whatever spell components, weapons, or other objects you need while making use of the ability. Spells are sometimes exceptions to the rules, but when there are no rules to be excepted they can be guidelines for how to run things.
As a GM, I would question other creatures Burrowing while armed, but the Troxs' extra limbs would be plenty of justification to allow it.