What you see here is the mushroom, basically the gonads of the organism (technically the “fruiting body”, something that is there to spread the spores). The bulk of the fungus is inside the dead tree, putting out digestive juices and eating it.

There’s no real harm to it to take it out. Or to use a knife and take a piece of it. It’s common practice to use porous bags to carry them so you’re helping it spread the spores around as you’re foraging.