What does Hebrews 13:10 mean?

Earlier passages noted that the physical components of the temple were really meant to symbolize the work of Jesus Christ. His suffering, on our behalf, was presented as a superior sacrifice (Hebrews 9:12), given in a superior place (Hebrews 9:24), with superior results (Hebrews 9:25–26). The writer also pointed out, extensively, Old Testament Scriptures predicting God's intent to provide this new covenant, rather than to rely on a system of animal sacrifice (Hebrews 8:6–8). At one point, this was related to the rights of a son as opposed to the rights of a slave (Hebrews 3:5–6).

This verse reflects some of that earlier imagery. Those who insist on pursuing salvation through the old covenant "have no right" to the benefits of the "altar" of the new covenant. This parallels a point made by Paul in the book of Galatians, where he points out that those who attempt to be saved through their works are "fallen away from grace" (Galatians 5:4). The two approaches—faith in Christ versus earning salvation by works—are not compatible (Romans 11:6). No matter how sincere they may be, those who put their trust in rituals, sacraments, or adherence to a moral code—those who "serve the tent"—are not justified before God.

Upcoming verses will continue to explain how Jesus' sacrifice is meant to symbolically fulfill the system of sacrifices offered under the old covenant. In doing so, the writer will continue to pull together various points and comparisons made throughout the letter.