One more word on the machlokes (Nazir 47) R' Eliezer and Chachamim regarding when no one else is available except a kohein and nazir, which should become tamei to bury a meis mitzvah. Both approaches suggested in the previous posts share the common denominator of approaching the Mishna as a debate over who ranks higher in the heirachy of kedusha, the kohein or the nazir. The debate is similar (to use R' Akiva Eiger's comparison) to the debate over which takes precedence, tadir or mekudash.
R' Yosef Engel suggests a radically different approach to the Mishna that is as equally brilliant as the others. Recall that R' Eliezer's argument in favor of the kohein becoming tamei stems from the fact that if a nazir becomes tamei he is penalized by having to offer a korban. We had understood that to mean that the kohein, who suffers no penalty, is heiarchichly on a lower level of kedusha than the nazir. Perhaps not, says R' Yosef Engel. Perhaps everyone agrees that the kohein ranks higher on the heirarchical scale of kedusha. However, R' Eliezer argues that since the nazir must offer a korban, it proves that he suffers more severly by the desecration of his kedusha. Should the kohein whose kedusha is greater avoid becoming defiled, or should the nazir, who despite having a lower kedusha suffers more greatly by its loss, avoid being defiled? That is the crux of the argument between R' Eliezer and Chachamim.