Should I be continually weighting the handlebars? When riding a non-technical flat or ascent, should the arms support some of the rider's weight?
I have noticed that even in those situations, I can freely let go of the bars and continue pedaling without shifting my weight - all the time being bent forward. Maybe this is because of my bike's geometry, my technique is wrong or this is just normal.
To answer the questions in the comments, I am not sitting up, nor am I strong. I feel as if 80% of my weight is on the saddle, 18% on the pedals (striving to make consistent circles) and 2% on the handlebars (there is some change in my posture when I let go the bars, but it's like 1cm at most). Flat pedals - haven't ridden clips for a while.

It appears that the stem has been replaced with 50mm shorter one, and the fork with 30mm longer one. With my saddle all the way up, the handlebars are still a little higher.
The question stands, however - is this (a) efficient and (b) healthy position for cycling level roads and climbs?
Yes, the arms should support some of a rider's weight.

However, outside of a hard effort, the hands should be light on the bars and the arms should be relaxed. I'm reminded of a brief quote from Wheelmen,


  "[Eddie] Borysewicz adjusted the [American] riders' bikes, taught them proper positioning, and described how to use their abdominal muscles to keep themselves steady on the bike and their hands light on the handlebars."


Making sure your hands and arms are loose helps to not only take pressure off your hands, but it also can help keep your shoulders and breathing more relaxed as well. Try riding with a tense upper body for a long period of time and you will see the difference.

Therefore, while it probably isn't feasible to do for long periods of time, you should be able to remove your hands from the bars while continuing in the same position you just were (leaning forward, but not sitting up).