The Devil's Kettle is a pothole in the rock where a small river disappears underground in the Judge C.R. Magney State Park in Minnesota. Would it be possible to trace the path of the drainage conduit by throwing in a small waterproof GPS tracker?
GPS units require a clear view of the sky to fix their position. Because unlike radio signals used for communications, once the GPS signal have bounced in a wall becomes useless for computing the receiver's position. Therefore, the GPS would loss fix shortly after entering the pothole.
The studies made in the area that determined the point at which the water reenters the river, claim that the flow of water keep objects at the bottom of the pothole, explaining why they never resurface. So there is a high chance that you wouldn't see your GPS ever again.
However, if your GPS were to resurface, it would fix its position again soon after regaining a clear view of the sky. So, if you recover it (or if it had the capability to transmit its position in real time), you won't see a record of the track followed by the device, you would see only the entry and exit point.
For such problems, similar to a car going trough a tunnel, the GPS is usually augmented with inertial navigation systems that can estimate the displacements and compute the position by dead reckoning from the last GPS fix. Despite that the turbulence within the water flow in the pothole would make very challenging to keep track of the trajectory of the device underground, a sophisticated enough device should be able to do it.