Jamieson, Fausset and Brown on Hebrews 13:10

10. Christianity and Judaism are so totally distinct, that "they who serve the (Jewish) tabernacle," have no right to eat our spiritual Gospel meat, namely, the Jewish priests, and those who follow their guidance in serving the ceremonial ordinance. He says, "serve the tabernacle," not "serve IN the tabernacle." Contrast with this servile worship ours.

an altar--the cross of Christ, whereon His body was offered. The Lord's table represents this altar, the cross; as the bread and wine represent the sacrifice offered on it. Our meat, which we by faith spiritually eat, is the flesh of Christ, in contrast to the typical ceremonial meats. The two cannot be combined (Ga 5:2). That not a literal eating of the sacrifice of Christ is meant in the Lord's Supper, but a spiritual is meant, appears from comparing He 13:9 with He 13:10, "with GRACE, NOT with MEATS."