I would assume that dropping flow rates or a drop in chamber pressure would cause the engine to stop thrusting, but I am unaware as to what specifically these conditions might be.
When thinking about what could happen, a fruitful method of approach can be considering what has happened.
This chart from the early days of SSME testing shows some of the ways things can go wrong.
Turbopump failures resulting in loss of flow and Other Bad Things
Burn-throughs of combustion machinery
- Ruptures of piping
- Valve failures