Don't know about those probes specifically, but of course it's possible to damage them. If you've got a FET probe, for example, you just need to get enough voltage to get the junction to breakdown and carry current, which anything outside the spec of the probe or high enough ESD would certainly damage it.
My guess is that you mean if regular circuit voltages could damage it if powered off.... and I guess that's a trickier one to answer unless you know the architecture of the probe. I doubt any manufacturer recommends it, though. Looking through the manual for a LeCroy HFP2500, the say not to leave it connected to a voltage source if it's not connected to a scope, but they don't say if the scope has to be on - could just be to prevent floating issues when reconnecting to a grounded scope.