My friend and I have recently started street biking, and we have a little disagreement - he thinks it is more work to sit down and pedal rather than stand up. I say standing up just uses more muscles of the body rather than isolating quads, but he says that standing up is easier because you're simply using gravity to push the pedals. I don't think that's consistent with physics, but I want to see if anyone has any conclusive explanation to end this disagreement.

If you are sitting more of the energy from your pumping is translated into the bike's drivetrain, and then into speed/climbing. The less the bike is wiggling back and forth the better. Wait til you get into clipless and you use the whole pedal stroke. Still upper body, pedal like hell..... welcome.

Quoting from https://www.sheldonbrown.com/standing.html:

"Standing pedaling allows you to apply more force to the pedals than is possible seated, because you can rest your entire weight on the driven pedal, and, even more, by pulling up on the handlebar, you can push the pedal with more than your actual weight...but is this a good thing?

Pedaling that hard is very stressful to the joints, and to the bicycle, and usually involves a level of effort that cannot be sustained aerobically. Unless you have unusually good form, it also tends to involve a fair amount of thrashing from side to side, which is a waste of energy. The added stress flexes many parts of the bicycle, and the energy required to do this flexing is not usually recovered when the parts straighten back out."

Additionally, you are quite right that more muscles are used because you are now supporting your body weight instead of the saddle doing this, so the heart rate for the same speed will tend to be a bit higher.

Tell him, by his standards every rider in the Tour De France pedals with the wrong method.

If your bike is badly set up, for example the seat is way too low, and you are using too high a gear, I suppose it's possible that your efforts would be so inefficient that you would cycle more effectively standing up, but otherwise seated is more efficient.
The problem with his gravity explanation is that while you use gravity to push down the pedal, at the same time you have to resist the effect of gravity pulling you down too. When you're seated, the bike supports part of your weight, while when you're standing, your leg muscles have to hold you up.

Ps. Invite him to go for a long ride with you and leave his seat at home. We'll see who lasts longest.

I sort of passed high school physics, and that doesn't make sense to me: the energy put into the system can't just be "lost", it either is there or is converted to another form. Unless your bike parts are heating up or giving off light, the energy is still there, no?. I think the difference between rigid and flexy parts and frames is the speed and/or "feel" of how the energy is fed back.

Some of the flex is lost as heat in the parts that are bending. The more bending you do, the more energy is lost through heat. However, this is a very miniiscule amount. A much larger proportion of the inefficient loss of power is in moving the soft fleshy human body around laterally in motions that's useless in moving the pedals down. Also on a single standing stroke you're using more of your body's mass, but before the very second stroke, you have to move that body mass back up in order to drive down the other pedal. You basically have to give back the extra push on the opposite side.

The real source of the extra power is that you get to use some of your strongest muscles at their most poweful part of their motion, near the limits of extension. So standing up lets you use several of your muscles to full contraction, giving you more power. However, making muscles exert near their maximum force is also inefficient, consuming more oxygen per watt generated than sitting down. You also drive them over their lactate-threshold easier, thus you won't be able to keep it up as long.

Sure you get to have gravity assist you when you stand, but it's not free energy. You have to lift your body at the top of each pedal stroke before you can recover the energy via lowering it again. I agree with Danno - as a bent rider I can't stand, and what I notice is that I can't lift my hips away from the pedals, which limits me in how much of the pedal stroke I can have my knees in a less-stressful (straighter) angle.