I own a condo and the heating is from a hot water radiator. We have a hot water radiator at the living room and there is quite a noise (some clanking but mostly like water running on a pipe) coming from it even though the main valve is all the way off. The one in the bath and bedroom make no noise on or off. Any tips appreciated.
It depends partly on your location but many radiator systems have bypass pipes through which water passes to and from other radiators on the same circuit - even it that particular radiator has its valve turned off.
Air in the heating pipework can cause the sort of noises you describe and can be eliminated by bleeding the air from the system.
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A clanking noise is often caused thermal expansion/contraction in the pipes feeding the radiators, as the slip in their mounting brackets. That noise can resonate through the pipes for quite some distance, and as @RedGrittyBrick pointed out turning off the radiator may not turn off the loop which feeds it.
The only cure I know for that involves figuring out where the twang is coming from and changing how the pipes are mounted at that point. Which is often more trouble than it's worth.
How recently did you move into the place? Odds are that after a year or two you'll stop hearing the "normal" noises of the place unless you're specifically listening for them.