38 Now this is that which thou shalt offer upon the altar; two lambs of the first year day by day continually. 39 The one lamb thou shalt offer in the morning; and the other lamb thou shalt offer at even: 40 And with the one lamb a tenth deal of flour mingled with the fourth part of an hin of beaten oil; and the fourth part of an hin of wine for a drink offering. 41 And the other lamb thou shalt offer at even, and shalt do thereto according to the meat offering of the morning, and according to the drink offering thereof, for a sweet savour, an offering made by fire unto the LORD.
42 This shall be a continual burnt offering throughout your generations at the door of the tabernacle of the congregation before the LORD: where I will meet you, to speak there unto thee. 43 And there I will meet with the children of Israel, and the tabernacle shall be sanctified by my glory. 44 And I will sanctify the tabernacle of the congregation, and the altar: I will sanctify also both Aaron and his sons, to minister to me in the priest's office. 45 And I will dwell among the children of Israel, and will be their God. 46 And they shall know that I am the LORD their God, that brought them forth out of the land of Egypt, that I may dwell among them: I am the LORD their God.
-- Exodus 29:38-46 (KJV)
תָּמִֽיד appears twice in this passage, the first in verse 38: לַיּ֖וֹם תָּמִֽיד = "day by day continually"; and the second in verse 42: תָּמִיד֙ לְדֹרֹ֣תֵיכֶ֔ם = "continual ... throughout your generations". It is pretty clear that this is what the writer of Hebrews was referring to in Hebrews 7:27.
This perpetual daily sacrifice is again mentioned in Numbers 28:1-8, with many mentions of, "beside the continual burnt offering" in regard to other particular offerings (Numbers 28:10,15,23,24,31; 29:6,11,16,19,22,25,28,31,34,38).
This perpetual daily offering is mentioned in 1 Chronicles 16:40 in reference to David bringing the Ark of the Covenant up to Jerusalem.
In Daniel chapter 8, the prophet speaks of the "Transgression of Desolation" when the daily sacrifice is taken away.
A number of comments have noted that this answer does not address the suggestion by the writer of Hebrews that the high priest was a participant in the sacrifice.
The point of the daily sacrifice was clearly the maintenance of the holiness of the Tabernacle of the congregation, the altar, those who ministered there, the high priest and the priests, and the children of Israel. This was a DAILY need because God said, "I will meet you, to speak there unto thee. And there I will meet with the children of Israel ..."
There can be little doubt that this perpetual daily sacrifice is what the writer of Hebrews was referring to, since there is nothing else that could possibly fit what he describes. So, each and every reader of Hebrews 7:27 is faced with the same dilemma as those who read Matthew 12:40, i.e. the LETTER of the Law vs. the SPIRIT of the Law. Some will decide one way, some will decide the other, and some won't give a fig.
God says concerning the daily sacrifice, "... and the tabernacle shall be sanctified by my glory.". So, the presence of the glory of the LORD was the evidence that He was at home in the Tabernacle. If the daily sacrifice ceased, then the presence of the glory of the LORD would cease also.
Who bore the responsibility for ensuring that the daily sacrifice did not cease? Surely, the high priest. So, securing the ongoing holiness of the Tabernacle of the congregation, the altar, those who ministered there, the high priest and the priests, and the children of Israel, AND the ongoing presence of the glory of the LORD, rested with high priest.
In this way, bearing the weighty responsibility that the daily sacrifices did not cease, the high priest offered up daily for his own sins and the sins of the people.
For those who are still puzzled, let me make it plainer: if you instruct someone to do something, and you take steps to ensure that he does it, then you are doing it.