I bought a 12 inch saucepan to cook with. Only when I brought it home did I remember that my largest (electric) burner is 8 inches in diameter. The label with the pan says to use it with a similar-sized burner. Will I cause harm to the pan or to my food if I use it with my 8 inch burner?
I do not think that you will cause harm to either the food or pan. The size of the burner isn’t all that critical to the size of the pan (though I cook on a gas stove where you adjust not only the temperature but the size of the flame). Generally, I think the size of the pan has more to do with the quantity of food being prepared and making sure that you provide sufficient space to evenly and adequately cook whatever you are making. BTW, always remember to preheat the pan before adding the food.
You shouldn’t have a problem if the pan is a good heat conductor. Copper is great but very pricy. Most pans are aluminum and stainless steel, many of which come with a thick bottom on the pan to better conduct the heat to the entire pan. If the bottom is thin, though, you will have issues with the pan being hotter where it touches the element, and therefore more prone to bruning / scorching.
I think the more important thing is just the opposite of what you are asking. Using a pan larger than the burner is okay, it’s using a pan smaller than the burner that is the problem. Then you risk burning yourself on the handle, starting a fire from the exposed element, etc. I would wager that that is a standard warning that goes on all cookware, regardless of it’s size, to limit the manufacturer’s liability.
Depending on the quality of the pan, food will cook unevenly due to hot and cold spots. It’s not a major deal, depending on what you want to do with it. If you want to sear something, keep it within the inner 8" of the pan.
To resolve some of the questions/concerns expressed in replies - do you want to identify which saucepan (maker and type) you bought?