The first time we see white clothing mentioned in the Bible is the High Priest’s garments on the Day of Atonement. [Lev 16; 23:26-32; Num 29:11 ] These passages never actually uses the word “white” however, the garments are made of plain linen with no dye mentioned. They were to represent perfect purity which was sought once a year in the special sacrifices of the Day of Atonement.

This is why Zechariah tell High Priest Joshua to take off his filthy garment.

3 Now Joshua was standing before the angel, clothed with filthy garments.  4 And the angel said to those who were standing before him, “Remove the filthy garments from him.” And to him he said, “Behold, I have taken your iniquity away from you, and I will clothe you with rich apparel.”  5 And I said, “Let them put a clean turban on his head.” So they put a clean turban on his head and clothed him with garments; and the angel of the LORD was standing by. Zech. 3:3-5 

Those who stand before or need God are described as wearing white.

And lo, six men came from the direction of the upper gate, which faces north, every man with his weapon for slaughter in his hand, and with them was a man clothed in linen, with a writing case at his side. And they went in and stood beside the bronze altar. Ezek. 9:2

I lifted up my eyes and looked, and behold, a man clothed in linen, whose loins were girded with gold of Uphaz. Dan. 10:5

And I said to the man clothed in linen, who was above the waters of the stream, “How long shall it be till the end of these wonders?” Dan. 12:6

The implication is that these are holy garments and one must wash before putting them on.

He shall put on the holy linen coat, and shall have the linen breeches on his body, be girded with the linen girdle, and wear the linen turban; these are the holy garments. He shall bathe his body in water, and then put them on. Lev. 16:4

I said to him, “Sir, you know.” And he said to me, “These are they who have come out of the great tribulation; they have washed their robes and made them white in the blood of the Lamb. Rev. 7:14

There is, however, some effort on the part of believers to obtain such spiritual clothing. One must conquer.

4 Yet you have still a few names in Sardis, people who have not soiled their garments; and they shall walk with me in white, for they are worthy. 5 He who conquers shall be clad thus in white garments, and I will not blot his name out of the book of life; I will confess his name before my Father and before his angels. Rev. 3:4-5