The Mishna, in Zevachim 2:1, lists a number of circumstances that can disqualify the essential services that korbanot require, and thus the korbanot themselves. Among these: if the kohen performing one of these services is "lacking garments," the associated korban is disqualified.
R' Ovadia miBartenura, channeling the Talmud (Zevachim 19a), explains that "lacking garments" includes if there is any foreign object separating either the kohen's tunic or his pants from his skin. The Talmud's source for this requirement of no interruption is Leviticus 6:3:
וְלָבַ֨שׁ הַכֹּהֵ֜ן מִדּ֣וֹ בַ֗ד וּמִֽכְנְסֵי־בַד֮ יִלְבַּ֣שׁ עַל־בְּשָׂרוֹ֒ וְהֵרִ֣ים אֶת־הַדֶּ֗שֶׁן אֲשֶׁ֨ר תֹּאכַ֥ל הָאֵ֛שׁ אֶת־הָעֹלָ֖ה עַל־הַמִּזְבֵּ֑חַ וְשָׂמ֕וֹ אֵ֖צֶל הַמִּזְבֵּֽחַ׃
The priest shall dress in linen raiment, with linen breeches next to his body; and he shall take up the ashes to which the fire has reduced the burnt offering on the altar and place them beside the altar.
This verse makes it clear that the pants must be in contact with the kohen's skin, but how does it tell us that that rule is true of the tunic, as well?