The Basis of the Forbidden Work of Dyeing
One of the thirty-nine works forbidden by Torah law on Shabbat is the forbidden work of dyeing. The Mishnah in Masechet Shabbat (73a) states likewise.
We have previously discussed that any work which was performed in the Mishkan (Tabernacle) was prohibited by the Torah to perform on Shabbat. One of the works commonly performed in the Mishkan was dyeing. This manifested itself in the dyeing of hides with various colors, marking animals with colors, and using the Techelet (blue) dye derived from the Chilazon.
The Talmud Yerushalmi (Shabbat 51b and commentaries ibid.) states that although the Jewish nation was in the desert, there were dyes that they brought with them from Egypt, such as the Techelet dye. Indeed, the Gemara states that there were salesmen who came to where the Jewish nation was in the desert in order to sell them merchandise. It is in this manner that the Jewish nation was able to engage in dying in the Mishkan in the desert and it is therefore one of the forbidden works on Shabbat.
Color/Dye which is not Permanent
The Torah only forbids coloring on Shabbat using permanent dye, such as ink on paper which stays there for a prolonged amount of time; however, the Torah does not prohibit coloring using non-permanent dye. Nevertheless, our Sages have decreed that it is forbidden to color on Shabbat using even non-permanent dye.
Polishing Shoes or Painting One’s House on Shabbat
Included in the forbidden work of dyeing is one who paints the walls of his house with plaster or paint on Shabbat. This is certainly a Torah prohibition, for this is a permanent kind of coloring.
Similarly, one may not polish his shoes on Shabbat with shoe polish, for this constitutes a Torah prohibition of dyeing on Shabbat (even if one polishes over a place that had already been colored or polished). If one smears the polish onto the shoe, some say that he has transgressed an additional prohibition of “smearing” on Shabbat. Even if the consistency of the polish is very thin and watery and thus does not constitute the Torah prohibition of smearing, there is nevertheless a rabbinic edict forbidding the smearing of any cream or polish on Shabbat, even if it is thin and easily-smeared by hand. (See Responsa Yabia Omer, Volume 4, Chapter 28.)
Summary: It is forbidden to color on Shabbat using either a permanent or non-permanent dye. For this reason, it is forbidden to paint one’s house or polish one’s shoes on Shabbat. It is likewise forbidden to color on paper and the like on Shabbat.