I think not. A priest represents the people before The God. It is a prophet that speaks on behalf of The God to people. One man can be both, but the roles do not mix. David was a prophet and a king, but not a priest. Samuel was a prophet and a priest but not a King. Apparently, Melchizedek was a priest and a king, but not a prophet. Elijah and Elisha were prophets but not priests or kings, as were othe prophets. Only Jesus, The Anointed One, The King of Kings, and Master (didaskalos), as well as The Eternal High Priest, has the three-fold ministry.
Note that Hebrews is for our benefit, but was addressed to the Jews of the diaspora, as said by Peter (2 Pet. 3:15). It is without doubt that it was meant to illustrate the meaning of their religion under the Law, which pictured the Messiah to come and fulfill it.
That's a little oversimplified. The priests did have teaching duties. In Leviticus 10:9, for instance, the LORD tells Aaron, "You are to teach the people of Israel all the statutes that the LORD has spoken to them by Moses." In Deuteronomy 33:10, it is said of the Levites, "They shall teach Jacob your rules and Israel your law; they shall put incense before you and whole burnt offerings on your altar." See also Malachi 2:7, "For the lips of a priest should guard knowledge, and people should seek instruction from his mouth, for he is the messenger of the LORD of hosts." Also, the priest would speak for God when he blessed Israel with the Aaronic Benediction (Numbers 6:22-27), putting the name of YHWH on the people.
Further, the priest would represent the people, yes, but he also acted for God, for the people's benefit, by offering up the sacrifices the LORD had appointed and provided for their forgiveness. And in that regard, I think there is some correspondence to what Christ has done for us as our great high priest.
A think a distinction can be made that the priests spoke for God "within the system," whereas the prophets often were speaking for God "outside the system."