Should I be continually weighting the handlebars?
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).