Relatively recent fossils, in which the original calcium carbonate ground
substance is still present, would still be susceptible to Byne's Disease.
It would probably be more difficult to detect visually in the early stages,
since the surfaces of many such fossils are already somewhat roughened,
eroded, chalky, etc., and much of the original pigmentation is gone.  But
there is no reason why such specimens should be immune.  It is still the
same chemical compound, so exposed to the same chemicals under the same
conditions, it should still react the same.  Older fossils, in which the
original ground substance has been replaced by harder, non-carbonate
minerals, (especially non-calcific non-carbonate minerals), would probably
not be susceptible to this kind of chemical decomposition.

Materials can easily be tested for susceptibility to Byne's, if you have a
small piece of a specimen that you don't mind sacrificing.  Just drop the
piece of material into any acidic solution.  If you see bubbles rising from
the piece of material, you are seeing the same reaction that occurs in
Byne's Disease, only in accelerated form.  If you use a very weak acid, like
vinegar, the rising bubbles will be very fine and not as obvious, but can
still be seen by close observation.  If you use a stronger acid solution
(something like 10% muriatic acid), the bubble production will be
immediately obvious.  If you use a very strong acid (like 100% muriatic
acid), and a substantial piece of shell, the bubble production may be so
violent as to cause the acid solution to splash out of the container, so
that is not recommended.