The title says it all: Do airline pilots leave the aircraft's weather radar on at all times during the flight and display the data on one or both of the Navigation Displays, or only if clouds/ weather are visible / expected? I know it should not be active on the ground due to possible radiation harm, so my question relates to the airborne phases climb, cruise, descent.
Not necessarily; it is the pilot's prerogative to use the weather radar as required. If the weather is clear and the pilot decides that the weather radar display is going to take up valuable space in display, he/she can fly without the weather radar on. Of course, in cases like night flying, it is better to have it on during the entire flight, but it is the pilot's decision.
The regulatory requirements are for the weather radar to be operational only if potentially dangerous conditions are expected en route or for night flying. From CFR §135.175 Airborne weather radar equipment requirements:
(b) No person may begin a flight under IFR or night VFR conditions when current weather reports indicate that thunderstorms, or other potentially hazardous weather conditions that can be detected with airborne weather radar equipment, may reasonably be expected along the route to be flown, unless the airborne weather radar equipment required by paragraph (a) of this section is in satisfactory operating condition.
European regulations are also similar. From JAR-OPS 1.670 Airborne weather radar equipment:
(a) An operator shall not operate:
(i) (1) A pressurised aeroplane; ...
unless it is equipped with airborne weather radar equipment whenever such an aeroplane is being operated at night or in instrument meteorological conditions in areas where thunderstorms or other potentially hazardous weather conditions, regarded as detectable with airborne weather radar, may be expected to exist along the route.
Consider things from a risk perspective. If the equipment is there, and it provides useful information for safety of flight, I leave it on. Setting the tilt down can provide orographic information, map water bodies, etc.
Also, I tend to run the electronics in the plane if there is no adverse consequence to doing so. Electronics tend to be adversely impacted by water, and heat dissipation tends to keep the electronics dryer.
Of course it is not operated on the ground, mostly because the radiation from the unit is harmful in a couple of ways to life, because it may interfere with electronics on the ground or nearby planes, and because it can cause problems with devices like a heart pacemaker.