Just as you support your torso with your arms when bent over to catch your breath, it is normal to have some of your weight rest on your arms, at least when using dropped bars or another well-designed, efficient arrangement.

The straight bars, often nearly as high or higher than the seat, on hybrid bikes so hopelessly retard your riding that it barely matters where you put your weight to minimize the damage they do to your breathing, back, aerodynamics and comfort. On the better-designed of mountain bikes, you will note that the handlebars are very much lower than the seat, and are far enough forward to promote the optimal 45 degree back posture. Get your handlebars as low as possible on a mountain bike that you can still manage a downhill, or change it if you won't be around hilly trails for a while.

Downhill bikes often have higher handlebars, only because they need to keep weight back for effective braking and more control over the ride -- on steep downhills. Although they can be fun to ride on, for efficiency, riser bars are poor for anything but DH riding, or maybe really slow and technical singletrack.

Many of the advantages of using dropped bars apply to setting your hybrid or mountain bike's bars to a height which promotes a similar effect.

Lowering your bars gives you more power, as jerking on your bars as you pedal will add to the torque generated, without the unwanted side effect of pulling the front wheel off of the ground. The lower, more distant handlebars also let you move your saddle forward with respect to the pedals, which puts it in a sprinty position.