Are there any creatures that will eat their own limbs or body parts to survive?
What you're describing is actually a well-documented occurrence in individual cells and single-celled organisms. The phenomenon is called autophagy, and Wikipedia has a page about it here. The basic gist is that when cellular components stop functioning or need to be removed for some other reason, the cell essentially digests those components and reuses their parts to make something else. Cells can also consume themselves to prevent the spread of disease or in the case of extreme starvation, when the extra energy might be needed to keep the organism alive.

Self-cannibalism on the macro scale (larger than single cells) is significantly rarer, but it can and does happen. For example, leopard geckos (and maybe some other lizard species? I'm not sure) regularly eat their own shed skin pieces to conserve energy and nutrients. Sea squirts (a type of marine invertebrates) literally digest their own brains as a part of their life cycle, and apparently, some crickets have been known to consume their own wings. Generally, though, unless the limb in question is already damaged and/or rendered unusable, it would be a waste of energy for its owner to consume it, as the energy spent repairing the damage would likely outweigh the energy gained from its consumption in the first place. That is why it's a relatively rare behavior in multicellular organisms.