βλέπετε τὸν Ἰσραὴλ κατὰ σάρκα· οὐχ οἱ ἐσθίοντες τὰς θυσίας κοινωνοὶ τοῦ θυσιαστηρίου εἰσίν (1 Corinthians 10:18 Nestle 1904)
It seems that κοινωνοὶ τοῦ θυσιαστηρίου is a genitive of association (Compare Rom. 8:17). Furthermore, the substantival phrase οἱ ἐσθίοντες is the subject here (not θυσιαστήριον). Additionally the context suggests that the eater consumes the meat, not the altar per se (1 Cor. 10:16-17). For a similar use of such altar terminology, see Heb. 13:10.
"Paul's line of reasoning was proceeding as follows. Christians who eat the bread at the Lord's Supper thereby express their solidarity with one another and with Christ. Likewise Jews who ate the meat of animals offered in the sacrifices of Judaism expressed their solidarity with one another and with God. Therefore Christians who eat the meat offered to pagan gods as part of pagan worship express their solidarity with pagans and with the pagan deities" (Dr. Thomas Constable's Expository Notes).
It seems clear that Paul's words apply to the Christian congregation; furthermore, they should be understood symbolically. In other words, those partaking of the wine in the cup and consuming the unleavened bread at the Lord's Evening Meal do not literally share a meal with God, as the doctrine of transubstantiation holds. To the contrary, those partaking on Nisan 14 profess the communion that they enjoy with God, Christ and other partakers. This meal points to the extramental Christian KOINWNIA that exists among those serving YHWH.