Carbon fiber composites have no fatigue limit, but that is not really the question to ask, since they do not behave like isotropic, hard (i.e., brittle) metals. Steel (titanium, too) has a fatigue limit, so fatigue failure can be avoided with it. As you say, aluminum has no fatigue limit and can thus be more difficult to predict when it might fail under a given cyclic loading. But being a matrix, carbon is less predictable in its failure modes under fatigue.
If a material has a fatigue limit or “endurance limit,” it means that if the cyclic stresses applied to it are below a certain magnitude (measured in units like pounds per square inch), the structure will not fail due to fatigue. Look at the graph; if the stresses on the steel structure depicted in the graph are below about 28ksi (kilopounds per square inch) – or 28,000psi, the steel will never fail due to fatigue (of course assuming no rusting or impact damage). The fatigue curve for titanium looks similar. But you can see that the fatigue curve for aluminum never flattens out, so even small cyclic loads, if they go on long enough, will eventually cause it to fail.
Carbon fiber composites can indeed fatigue, but they do so very differently from metals. Metals tend to be isotropic (uniform in any direction), even though they can be mildly anisotropic in that they can have a predominant grain direction and, like wood, can be stronger along the grain than across it. (According to the coronial report you sent, this particular aluminum steering tube was not uniform in that it had an “inclusion flaw” in it.) By contrast, carbon fiber composites are extremely anisotropic, depending on how the layers of carbon fabric (usually unidirectional fibers) are laid across each other. And this can vary from fork to fork in the same batch, since the fabric pieces are laid up by hand.
Metals tend to eventually fail under fatigue at a single crack. Carbon fiber composites do not do this — they tend to degrade under fatigue throughout the entire volume of the structure. Composite materials fail due to fatigue in four basic ways: cracking of the matrix (i.e., the resin that holds the fibers together), delamination (peeling apart of one layer of fabric from another), breakage of fibers, and debonding of individual fibers from the resin. You may be able to hear delamination, debonding, and cracking by tapping a coin along the fork; where the matrix is intact, it will make a nice “clack” sound, whereas the sound will be deadened if layers have peeled away from each other or if fibers are cracked or no longer stuck together. Cracking of the fibers and/or of the matrix can be deep within the layers and may not be visible from the surface. Fatigue cracks in metal structures always propagate from the surface (the trick is finding the cracks when they are small).
Long answer to your question, but carbon forks can be subject to fatigue. However, in the absence of crashes or impacts, high-quality carbon forks tend to be highly resistant to fatigue. If your fork goes through a hard crash or receives a hard impact, you ought to replace it right away. In the absence of crashes and/or impact damage, it still makes sense to replace full-carbon forks every now and then. Unless you weigh over 200 pounds, however, you can probably figure a good fork will last a long time (10 years or so?) before it needs replacing. The heavier you are, the shorter the interval between replacements. A sub-130-pound rider can probably assume a full-carbon fork will last them a lifetime unless they go through a crash or suffer an impact.
The issue of course is how a bike is used over the years, since 10 years of use for different riders will be different mileage and riding surfaces and tire pressures. To truly monitor fatigue, you would need have an odometer on it and a way to measure the magnitude of the stresses it receives. I don’t see this appearing on a smartphone app anytime soon.