Sacrificing and eating denoted fellowship, a oneness with those who were taking part in the sacrificing in identification with the object of sacrifice, and thus believers cannot be part of pagan communions.
And as the next chapter (1Cor. 11) teaches, to take part in the Lord’s supper is to commemorate it by care for each others, which the love feast, the “feast of charity,” expressed.
And thus to not do so, but to take hypocritically part in it selfishly, as they were in 1Cor. 11, is “not to eat the Lord’s supper,” as they were not recognizing the neglected members as part of the body of Christ, this being the context, and which follows into the next chapter.
What 1 Cor. 11 teaches on this is more fully shown here: http://www.peacebyjesus.net/Bible/1Cor._11.html#11
1 Corinthians 10
14
Therefore, my beloved, shun the worship of idols.
15
I speak as to sensible men; judge for yourselves what I say.
16
The cup of blessing which we bless, is it not a participation in the blood of Christ? The bread which we break, is it not a participation in the body of Christ?
17
Because there is one bread, we who are many are one body, for we all partake of the one bread.
18
Consider the people of Israel; are not those who eat the sacrifices partners in the altar?
19
What do I imply then? That food offered to idols is anything, or that an idol is anything?
20
No, I imply that what pagans sacrifice they offer to demons and not to God. I do not want you to be partners with demons.
21
You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons.