Is there a problem with wearing tefillin whose paint has started to chip off? Would such a situation require one to get the tefillin repainted? Can someone still be yotzei by tefillin whose paint is chipped?
It is a halacha l'Moshe miSinai that the r'tzu'os must be black (M'nachos 35a), and this is a requirement for valid t'fillin. The straps must be re-blackened if the paint becomes scratched or abraded (Mishna B'rura 33:19). Abrasions are especially common (if not readily noticeable) in the vicinity of the knot of the shel yad, and special attention should be paid to that area of the r'tzu'a (Bach OC 32:35).
The Bi'ur Halacha (to 33:3, s.v. הרצועות שחורות) is undecided regarding whether the entire length of the straps must be black for the t'fillin to be valid, or whether the t'fillin are valid if even the minimum required length for r'tzu'os are black (see Shulchan Aruch OC 27:8,11 for the minimum required lengths). Rabbi Kaganoff cites R' Yosef Chaim Sonnenfeld as ruling that if even the majority of the r'tzu'os are black, the t'fillin remain valid. (This may refer to a contiguous length of the r'tzu'os, possibly beginning from the top down).
Although most poskim maintain that it is also a mitzvah for the batim to be black (see for example Shulchan Aruch OC 32:40), there is disagreement regarding whether they must be black for the t'fillin to be valid. The Mishna B'rura (32:184) cites multiple acharonim who rule stringently but say that a person may rely on the lenient opinions when t'fillin with completely black batim are not available.
According to the Shulchan Aruch (OC 33:4), it is preferable to paint the t'fillin lishmah, while the Rama (ad loc.) holds that the straps must be painted lishmah and that it is proper to paint the batim lishmah. Since the Rama does not hold that lack of paint on the batim invalidates the t'fillin, he writes that a lack of lishmah does not invalidate the painting of the batim.