Burrowers move underground; each turn they emerge and attack a 3-wide area in front of them. On taking any damage (weapon damage, fire, pushing a unit into them, etc.) they hide underground for the rest of the turn. They are stable, and therefore unable to be pushed. This combination makes them fairly difficult to kill quickly.
Burrowers often require whittling them down over several turns or taking them out in one or two strong attacks.
Burrowers can be instantly killed by water that appears on a tile they are occupying. For example, a burrower can be killed from a tidal wave on Archive, or by the water created when the dam is destroyed. However, as they have the stable attribute and can't be moved, this requires a degree of luck.
Fire will damage them for 1 damage, causing them to crawl back right before their attack. But they do not remain on fire when they re-emerge next turn. (Presumably they will never emerge on a fire tile unless there are no other options).
(To test: does water block their movement underground? Presumably above-ground mechs do not block movement.)