If a Kohen is prohibited from becoming Tamei, and people become Tamei upon dying, then wouldn't it be prohibited for a Kohen to die?
A kohen is allowed to become tame to a close relative, and adam karov etzel atzmo, a person is his own relative, so it's all right. A kohen gadol, on the other hand, who cannot become tame even to a relative, cannot die unless he's a mes mitzva, someone who dies where there's no one around to bury him. (A kohen gadol can become tame to a mes mitzva.) That's why so many kohanim g'dolim died in the kodesh kadashim during the second Temple, so there'd be no one around when they died.
The question is based on the premise that dead people are tamei. However, this premise is questionable.
The Gemara (Nidda 70b) discusses the taharah status of a once dead person after reincarnation:
מתים לעתיד לבא צריכין הזאה שלישי ושביעי או אין צריכין
The simplest explanation of this shaila is whether dead people become impure or not. We know that someone who touches a corpse becomes impure. Is the corpse itself impure?
The Sifri (Parshas Chukas) darshens the following:
כל הנוגע - נוגע במת טמא, אין מת עצמו טמא
It seems that only one who touches the body is rendered impure, not the body itself.
[However, see Rashi in Pesachim 18a that we consider everything to be touching itself; אין לך נגיעה גדולה מזו. This would contradict the above Sifri, which sounds like the corpse itself doesn't become impure. See also Chasam Sofer YD 337]