Is it biologically feasible for an animal to evolve in a way that allows the body to produce carbon nanotubes? How could this happen? What advantages would it give?
I think the question "Is it biologically feasible for an animal to evolve with carbon nanotubes?" is easier to answer if you mean "animals produce carbon nanotubes" similar to the way known animals produce bone, or tendons, or chitin, etc. (This is as opposed to animals evolving fundamentally from nanotubes, in the way we evolved from amino acids; since that is fairly inconceivable.)
The answer here still cannot be hard science, since there are no known biological means by which to make carbon nanotubes. But to offer you ANY answer, I'll note that biology is a game of permutations, where the random variations of the behaviors of molecules, proteins, systems, etc. are naturally selected-for, as environments change, bringing new selective pressures to bear. In concept, perhaps an excretory process that rids an organism of an excess substance (like calcium carbonate in ocean creatures) starts having some beneficial side-effect (like a protective layer).
So, if you really want deep exposition to back-up your built-world, then you need these two aspects: (1) some sniff-test-passing (i.e., non-absurd) mode for carbon to be either excreted, or otherwise be a useful building block already used biologically for other uses, and (2) some environmental shift that made the nanaotubes useful, and thereby selected-for. Perhaps for their structural strength, or their electrical or thermal conductivity. Marry this environmental pressure to what you have already decided your built world contains, as the defining challenges for its denizens, and it will make sufficient sense to your readers, or game players, whichever they are.
Plants opted to use CO2 as a means of storing energy and producing structural materials in Earth's deep history, making sugars, starches, and fibers. Perhaps in a world lean on oxygen, they might lack the carbohydrate chemistry spectrum, and so would eventually instead come upon carbon nanotubes. Zoobiologists can explain why a world lacking carbohydrate chemistry fodder (like abundant oxygen) is unlikely to develop life (as we know it), and I'm inclined to believe them. But you are building a fictional world, so you have to embrace fictional undergirding.
Biological reactions typically involve organic compounds, in other words mixture of elements of which one is usually carbon.
Carbon nanotubes are made of pure carbon arranged in a specific way. The thermodynamic of the pure carbon with the free energy involved makes it hardly capable of being produced in a biologically compatible way. In layman words, carbon is too eager to react with something else or in other forms than to yield and make a nanotube within a living organism.
Moreover I am not sure carbon nanotubes would give any advantages when we weight the effort and the benefits within the constrains of a living organism: they have a formidable tensile strength along their axes, but a living organism doesn't have the luxury of choosing how to load its body. I think therefore that a body part which can be amazingly resistant only when loaded in a certain, very specific direction and would be lousy when loaded otherwise would not give a real advantage.