When Phobos meets its predicted doom in a few 10s of millions of years, whether it breaks up or collides with the surface of Mars, will this change have any noticeable effects (permanent or temporary) on the Earth?
No. Phobos is small - just 11 Km across - the size of a small city. Mars (and Phobos) is so far away that a Phobos impact will not affect Earth much. (Mars ranges from 100 billion meters away to nearly 400 billion meters depending on its and Earth's position in their orbits.)
When Phobos hits the Roche limit as it will break up and become a thin light-grey planetary ring around Mars for a few million years.
When it breaks up, (almost) all the debris should stay near Mars because the escape velocity from its orbit around Mars is greater than the velocity imparted to boulders of Phobos as it self-destructs.