I'm having a little trouble trying to wrap my mind around what Hebrews 13:10-13 is teaching. What is your understanding of these verses? I really do value your observations and commentary.
I'm going to expand the range just a little bit so we get the complete thought:
"Do not be carried about with various and strange doctrines. For it is good that the heart be established by grace, not with foods which have not profited those who have been occupied with them. We have an altar from which those who serve the tabernacle have no right to eat. For the bodies of those animals, whose blood is brought into the sanctuary by the high priest for sin, are burned outside the camp. Therefore Jesus also, that He might sanctify the people with His own blood, suffered outside the gate. Therefore let us go forth to Him, outside the camp, bearing His reproach. For here we have no continuing city, but we seek the one to come" (Hebrews 13:9-14).
Keep in mind that the letter to the Hebrews was targeted to Christians from a Jewish background. From the time the church was established, there were elements who constantly attempted to push Christianity back toward the Old Law. "But some of the sect of the Pharisees who believed rose up, saying, "It is necessary to circumcise them, and to command them to keep the law of Moses"" (Acts 15:5) is just one example. Paul dealt with this issue at length in his letter to the Galatian churches.
"I marvel that you are turning away so soon from Him who called you in the grace of Christ, to a different gospel, which is not another; but there are some who trouble you and want to pervert the gospel of Christ. But even if we, or an angel from heaven, preach any other gospel to you than what we have preached to you, let him be accursed. As we have said before, so now I say again, if anyone preaches any other gospel to you than what you have received, let him be accursed. For do I now persuade men, or God? Or do I seek to please men? For if I still pleased men, I would not be a bondservant of Christ" (Galatians 1:6-10).
Some there were being pulled away by Judaizing teachers. "O foolish Galatians! Who has bewitched you that you should not obey the truth, before whose eyes Jesus Christ was clearly portrayed among you as crucified? This only I want to learn from you: Did you receive the Spirit by the works of the law, or by the hearing of faith? Are you so foolish? Having begun in the Spirit, are you now being made perfect by the flesh? Have you suffered so many things in vain--if indeed it was in vain?" (Galatians 3:1-4).
Among the various laws being thrust upon Christians were the dietary laws from the Old Testament. "So let no one judge you in food or in drink, or regarding a festival or a new moon or sabbaths, which are a shadow of things to come, but the substance is of Christ" (Colossians 2:17-18). It is to these that the writer of Hebrews is alluding. There were some attempting focused on what foods to eat thinking that these made them righteous, and, in doing so, lost track that righteousness comes by the grace of God and not keeping the Old Law. "Where is boasting then? It is excluded. By what law? Of works? No, but by the law of faith. Therefore we conclude that a man is justified by faith apart from the deeds of the law" (Romans 3:27-28).
Starting with Hebrews 13:10, the writer lapses into figurative language, drawing on images from the Old Testament that these false teachers wanted so strongly to impose. In the Old Testament, meat cooked on the altar could not be eaten by every Jew. Only the Levite priests could eat some of what was cooked there. "This shall be yours of the most holy things reserved from the fire: every offering of theirs, every grain offering and every sin offering and every trespass offering which they render to Me, shall be most holy for you and your sons. In a most holy place you shall eat it; every male shall eat it. It shall be holy to you" (Numbers 18:9-10). Using that imagery, the writer says that Christians have a source of spiritual food which the false teachers, even the priests under the Old Law, could not have. By using "tabernacle" instead of "temple" the writer further emphasizes that the old system was temporary, like a tent, and not permanent like a building. The Jews who attempt to maintain the old system have no right to the sacrifice of Christ. "Indeed I, Paul, say to you that if you become circumcised, Christ will profit you nothing. And I testify again to every man who becomes circumcised that he is a debtor to keep the whole law. You have become estranged from Christ, you who attempt to be justified by law; you have fallen from grace" (Galatians 5:2-4).
That Jesus' death could be compared to such sacrifices is argued in Hebrews 13:11-12. The sin offerings were not fully burned on the altar. Instead, the excess was taken outside the camp and burnt. "The bull for the sin offering and the goat for the sin offering, whose blood was brought in to make atonement in the Holy Place, shall be carried outside the camp. And they shall burn in the fire their skins, their flesh, and their offal" (Leviticus 16:27). By no coincidence, Jesus died on cross outside the city boundaries (John 19:17-18). In this way, he became a type of burnt offering; that is, a sacrifice in the category of burnt offerings. "And walk in love, as Christ also has loved us and given Himself for us, an offering and a sacrifice to God for a sweet-smelling aroma" (Ephesians 5:2). Jesus' sacrifice was the perfect offering for sin.
But because it was done outside the camp, it wasn't fully done as part of the Old Law. Thus, Jesus' blood reaches to his people, which includes those who were not Jews by birth.
"Therefore remember that you, once Gentiles in the flesh--who are called Uncircumcision by what is called the Circumcision made in the flesh by hands-- that at that time you were without Christ, being aliens from the commonwealth of Israel and strangers from the covenants of promise, having no hope and without God in the world. But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. For He Himself is our peace, who has made both one, and has broken down the middle wall of separation, having abolished in His flesh the enmity, that is, the law of commandments contained in ordinances, so as to create in Himself one new man from the two, thus making peace, and that He might reconcile them both to God in one body through the cross, thereby putting to death the enmity. And He came and preached peace to you who were afar off and to those who were near. For through Him we both have access by one Spirit to the Father" (Ephesians 2:11-18).
That is why those trying to remain under the Old Law had no right to partake of this ultimate sacrifice. To do so, they have to go outside the camp; they must come out of the Old Law. "Therefore, my brethren, you also have become dead to the law through the body of Christ, that you may be married to another -- to Him who was raised from the dead, that we should bear fruit to God" (Romans 7:4). The writer is inviting those who stubbornly hold onto the past, to let go and join themselves to Christ.
It is necessary, but Jerusalem won't survive, just as Christ foretold. "And Jesus said to them, "Do you not see all these things? Assuredly, I say to you, not one stone shall be left here upon another, that shall not be thrown down"" (Matthew 24:2). There is nothing left for the Jews under the Old Law, but there is a promise of a greater Jerusalem, a heavenly city, under the law of Christ. "For our citizenship is in heaven, from which we also eagerly wait for the Savior, the Lord Jesus Christ, who will transform our lowly body that it may be conformed to His glorious body, according to the working by which He is able even to subdue all things to Himself" (Philippians 3:20-21).