For example: can I turn a key on the other side of a solid door? Can I move objects that are behind me? Can I shut down a tank engine by turning an ignition key from outside a tank? Etc.

I would rule that you need to percieve the object you wish to manipulate. Meaning that you must either see the object or percieve it through psychic means. Otherwise you won´t know what to manipulate.

It woul be strange if a psyker could just say "I know there is a key somewhere inside this tank, I will turn it.", the psyker still would not know where that key is located and as such can not turn it.

I would say yes, you need line of sight, or at least some similar way to get specific and accurate information on what you are wanting to manipulate. A key on the other side of the door might be ok, as you can see there is a key in the lock (as long as you can see through the keyhole normally) and so know where it is.

Definitely no speculative use of the power. For example no "If there is a button on the other side of this wall I will press it!". Also I would probably not allow things where you can't get specific knowledge there and then, even if you have had previous knowledge of something, so no "Well, last time the starter button on that tank was there so I do that again using that knowledge even though I can't see it this time."

Basically 90% of the time I would say yes you need line of sight and need to know where it is (ie no doing it through security cameras unless you also know where the feed is coming from in relation to yourself. Basically, as it is Precision Telekinesis you need to know precisely where something is before you can manipulate it.