O.C. 610:4 says that it has become a common minhag to place "pleasant" coverings in shul (he doesn't specify a color, though) on Yom Kippur.
Mishnah Berurah commentary #9 says that in explaining Isaiah 58 that says "For the holiness of G-d's honored" that this is specifically a reference to Yom Kippur. Since the verse mentions "Shabbat", we usually honor Shabbat by eating and drinking. However, as we cannot do this on Yom Kippur, we instead honor the day by covering the tables in clean cloths. (I assume that the term "clean" is synonymous with white, however, I guess one could, technically use any color that looks "clean".)
This site mentions:
As far as I know, the first to mention these customs is R. Ephraim
Zalman Margaliot (Sha’arei Efrayim, Galicia, d. 1820), who also
provides the most detailed description of these customs:
The ritual objects — ark curtain, Torah mantles and table covers —
should be beautiful and splendid, to honor and glorify the Torah. It
is customary to make these in many different colors.
In honor of the High Holy Days, white ark curtains, Torah mantles and
table covers should be designated, for these are the days of judgment,
as is written (Isaiah 1: 18): “If your sins be like scarlet, they
shall be white as snow”. It was the custom to hang a white ark curtain
and cover the Torah scrolls used for the day’s reading in white