In Hebrews 13:9 Paul told his readers to refrain from various and strange doctrines. Don’t be carried about by them, or, in other words, don’t allow them to rule one’s life. We are the servants of the one we obey (cp. Luke 16:13; Romans 6:16). In the context of Paul’s epistle, believers can be deluded into serving a doctrine rather than Christ. Christ, himself, told the Pharisees that the doctrines of men, which they practiced and taught others to practice, led those who obeyed them away from God (Mark 7:6-13), and this was what Paul had in mind here.
We must not think that Paul’s statement in Hebrews 13:9 is disconnected from Hebrews 13:1-8. Everything fits together and is defined in the context of his other statements. Although Paul doesn’t elaborate on what the various and strange doctrines are, we do know that those who practice them had not been confirmed or established in the faith through mercy and kindness (grace; Hebrews 13:10). Nor, are these doctrines new, because Paul claimed that those who practiced them were never profited through their use. So, whatever Paul meant by various and strange doctrines, the Jews had practiced such things for some time, and it was apparent that they weren’t benefited by them.
What I believe Paul had in mind was the traditions (teachings, doctrines) of the elders, which were practiced by Jewish society at large (Mark 7:1-6). Jesus, of course, rebuked the teachers of such doctrines, saying their practice led people away from God (Mark 7:6-13). Paul had earlier commanded believers that they take measures to prevent others from judging their behavior through religious practices which may foreshadow Christ, but aren’t in themselves necessary to the Body of Christ. In other words, religious humility tends to puff up one’s fleshy mind, and obeying these various and strange doctrines (touch not; taste not; handle not) is nothing more than doctrines of men that require will worship and / or the worship of the messenger who taught them (cp. Colossians 2:16-23). Such doctrines would prevent the Jew from receiving the stranger in brotherly love (Hebrews 13:1-2) or identifying with the one, who was persecuted and ostracized by Jewish religious society (Hebrews 13:3). They also prevented the Jewish believer from accepting his gentile brethren as real and equal citizens in the Body of Christ (Hebrews 13:4). Such doctrines can’t serve Christ and must not be practiced by one who follows the Lord.
At this point Paul pointed to a third altar, concerning which the Levitical priesthood had no right to eat (Hebrews 13:10). Paul alluded to the importance of this alter by saying go forth to him who is outside the camp (Hebrews 13:13), i.e. outside the camp of Jewish society (or outside the world as this applies to the gentile believer). Just as Jesus was sacrificed (crucified) outside the gate of Jerusalem, outside Jewish society, as it were, so, too, it behooved the Jewish follower of Christ to bear his reproach by identifying with him rather than the common Jewish religious practices of that day (Hebrews 13:12).
What Paul had in mind in mentioning the “third altar” is alluded to in the Law of Moses (Hebrews 13:10-11). It wasn’t constructed inside the Temple compound. Rather, it was the altar that was constructed outside the Eastern Gate of the Temple, where the red heifer was slain and sacrificed as a whole burnt offering. Its ashes were what sanctified the Temple and all its contents and anyone who was ceremonially unclean (Numbers 19:1-9).
Ezekiel also referred to this place outside the camp in Ezekiel 43:21. Most translators, however, seem to imply this was the altar just outside the Temple, but this is impossible. Notice that the Altar of Burnt Offering was measured and in the process of being purified (Ezekiel 43:13-20). A bull was taken to a designated place (not the Altar of Burnt Offering) outside the Sanctuary (Ezekiel 43:21). The purifying process took seven days (Ezekiel 43:22-26), and only after that period of time were the priests permitted to offer sacrifices upon the Altar of Burnt Offering. Therefore, the altar where the bull was sacrificed and whose blood was used to purify the Altar of Burnt Offering had to be located elsewhere, outside the Sanctuary, and such things as these foreshadowed Christ’s own sacrifice. In fact he was crucified not far from this altar on Mount Olivet, and Paul told his Jewish brethren that those Jews who serve the altar in the Temple have no right to eat of the sacrifice that was slain on this third altar. Nevertheless, they who serve Christ have a right to partake of him, and they do so as long as they receive strangers as their brethren and recognize that he who is begotten in Christ is an equal heir, no matter what his nationality. All are one in Christ (Galatians 3:28).
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A Heifer is a female who has not borne a calf. After a heifer gives birth she is called a cow. An adult male is called a bull.