THE first indication of an oil problem is a run-away prop on
a single-engine with a typical constant speed prop. The
prop governor is just a high pressure oil pump and control
valve that must be supplied with engine oil under pressure
and with adequate volume. If the engine oil galleries and
lines are not full of oil, the governor will not work. On a
multiengine airplane, the prop will feather.
The prop, on a single, becomes fixed pitch and the throttle
controls rpm just like it was a solid fixed pitch. The
engine will run just fine as long as it has oil, but a
ruptured line or gasket will dump the oil and you don't know
exactly what is happening, so land ASAP.

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Most, but certainly not all, hydraulically controlled constant speed props
will fail to their finest pitch position if oil pressure is lost. There
have been exceptions--there was a high performance kit plane (I forgot the
name) which had the prop spring loaded the other way, such that any governor
failure would leave the porp in coarse pitch.

I have no idea how any of the auto feathering props on turbo-prop engines
really work, and therefore I don't know whether they would require oil
pressure to feather. However, I have not personally heard of any feathering
prop on any reciprocating engine which does not require oil pressure to
feather.

Peter