I broke a rear spoke and was riding home trying to baby the back wheel, especially on bumps. One of the things I did was stand up on the bumps, with the reasoning that taking weight off the seat would allow the back of the bike to bounce up more freely, thus reducing stress on the wheel. But then I began to think about the issue more. Me standing on the pedals does not reduce the amount of weight on the wheels, so does that really help with the load and shock of hitting a bump?

I would think that the standing on the pedals would shift the center of gravity is on the bike from between the pedals and rear wheel axle to at the pedals, so this would effectively change the force on each wheel.

Depends more on where your body is situated. E.G., Standing, but leaning back = more weight on the rear wheel than when sitting. Standing while leaning forwards= less weight on rear wheel than while seated.

Yeah, standing out of the saddle can move your center of gravity forward, because you will move forward to stand up.

The other benefit is that your body weight will be entirely suspended on the "shock absorbers" that are your knees. This will give a little compliance and reduce the impulse to the frame and wheels on each bump. Sitting on the saddle, you lose most of this compliance.

Standing up definitely reduces the load on the back wheel going over bumps. Your ankles, knees, hips, and spin all act like shock absorbing springs, bouncing with the bike. Since most of the mass on the bike is your body, you want to cushion the bike from your body. Like if you jump down a curb, you would want to be standing up, and then try to keep your body falling a bit longer than the bike to kind of spread out the energy over more time. It's also less painful. Sometimes I see people sitting bolt upright with their weight on the seat like a sack of potatoes. Every bump probably hurts their body as much as it hurts their wheels.