It's not very controversial that the Passover is replaced in the New Covenant by the Eucharist (whose names comes from that the Lord 'gave thanks'—ευχαριστησας—at His last supper—presumably a relatively long prayer which may validly be taken to account for what are later called 'liturgies:' what Jesus needs a few divine words to say, we are to be all the more grateful, and lengthy in our thanks, praise and petition). As the Passover commemorated freedom in the time of slavary to Egypt, and freedom from the death coming upon them, by the blood of a lamb marking their home, so the Eucharist commemorates the freedom of Christians from slavery to sin and the death coming upon the unrighteous, in the blood of the Lamb. Jesus invites them to a Passover as usually, but this time, all speak of a Lamb is left out, and instead the new Melchizedek (Heb 7:3b) makes use of the bread and the wine (Gn 14:18) and radically identifies Himself, the Lamb of God, as the new Passover (1 Cor 5:7).
As such, no distinction is to be made between the Passover 'table,' at which Jews sat to eat the sacrificial meal, and the Christians' 'table,' who 'partake of' (1 Cor 10:16; Cf. Ex 12:8) the body and blood of "the Lamb of God who takest away the sin of the world" (Jn 1:29).
The Table of the Showbread
This can't account for our reference here to 'table of the Lord,' since it is most clearly a reference to a table of sacrifice, the Table par excellence, God's altar, which, if St. Paul's parallels are to make sense, are in direct contrast to the altars to false gods, and the profitless sacrifices offered thereon.
Here, though, I think we can gather from its typological significance, a further symbol of the Last Supper, in that, just as this contained bread, 12 pieces of bread symbolizing the 12 apostles, wine was drank as the bread was consumed by the priest—once every week, just as is the custom for the weekly gathering of Christians for the Eucharist.
Another significant feature is that only the priest partook of the bread, symbolizing that Jesus made the twelve a kind of priesthood, now offering His Memorial (Lev 24:5-8) (the word 'priest' is derived directly from the word 'presbyter'). Cf. Mt 12:3-4. I find it significant that Jesus, before He had given the discourse in John 6, performed the miracle of the loaves and fishes (bread, flesh; manna, quail), and had the twelve distribute, almost in preparation for what they will be told on the night before His betrayal: "Do this for a commemoration of me" (as well noted by many authors of various stripes, this phrase contains sacrifical language used in the Old Testament).
The types and shadows are not exclusive, but speak to different aspects of the New Covenant fulfillment.
The Table of the Lord — the Altar of Sacrifice
Protestants and others usually mean a 'supper' when they say or even read "The Table" of the Lord, as in the Last Supper. (Vestigial concepts remain nontheless, such as speak of the 'altar call'). Except, in the context in which St. Paul was writing, "the table of the Lord," especially when used in the context of sacrifice, is Old Testament terminology for the altar of sacrifice.
Malachi 1:10-14 (DRB)
Who is there among you, that will shut the doors, [that you kindle no more the fire on my altar, aimlessly?] I have no pleasure in you, saith the Lord of hosts: and I will not receive a gift of your hand. 11 For from the rising of the sun even to the going down, my name is great among the Gentiles, and in every place there is sacrifice, and there is offered to my name a clean oblation: for my name is great among the Gentiles, saith the Lord of hosts. 12 And you have profaned it in that you say: The table of the Lord is defiled: and that which is laid thereupon is contemptible with the fire that devoureth it. 13 And you have said: Behold of our labour, and you puffed it away, saith the Lord of hosts, and you brought in of rapine the lame, and the sick, and brought in an offering: shall I accept it at your hands, saith the Lord? 14 Cursed is the deceitful man that hath in his flock a male, and making a vow offereth in sacrifice that which is feeble to the Lord: for I am a great King, saith the Lord of hosts, and my name is dreadful among the Gentiles.
Early Christianity was unanimous in seeing this as a reference to the New Covenant sacrament of the Eucharist—even as early as the Didache (A.D. 70-90), the earliest known Christian teaching document/catechism outside the New Testament:
But every Lord’s day do ye gather yourselves together, and break bread, and give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. But let no one that is at variance with his fellow come together with you, until they be reconciled, that your sacrifice may not be profaned. For this is that which was spoken by the Lord: In every place and time offer to me a pure sacrifice; for I am a great King, saith the Lord, and my name is wonderful among the nations.
(Notice the possible reference to St. Paul when it says to 'confess your sins beforehand' partaking: "but let a man examine himself first, and so let eat of that bread, and drink of that cup" (1 Cor 11:28).)
This is of course a paraphrastic, retrospective reference to the passage in Malachi quoted above.
Cf. Apostolic Constitutions (A.D. 70-120), Book VII, cap. 30; Book VI, cap. 23.
An important early Christian apologist, Justin Martyr, writes (A.D. 150-160):
God speaks by the mouth of Malachi, one of the twelve [prophets], as I said before, about the sacrifices at that time presented by you: ‘I have no pleasure in you, saith the Lord; and I will not accept your sacrifices at your hands: for, from the rising of the sun unto the going down of the same, My name has been glorified among the Gentiles, and in every place incense is offered to My name, and a pure offering: for My name is great among the Gentiles, saith the Lord: but ye profane it.’ [So] He then speaks of those Gentiles, namely us, who in every place offer sacrifices to Him, i.e., the bread of the Eucharist, and also the cup of the Eucharist, affirming both that we glorify His name, and that you profane [it]. (Dialogue with Trypho, a Jew, 41)
Accordingly, God, anticipating all the sacrifices which we offer through this name, and which Jesus the Christ enjoined us to offer, i.e., in the Eucharist of the bread and the cup, and which are presented by Christians in all places throughout the world, bears witness that they are well-pleasing to Him. But He utterly rejects those presented by you and by those priests of yours, saying, ‘And I will not accept your sacrifices at your hands; for from the rising of the sun to its setting my name is glorified among the Gentiles (He says); but ye profane it.’ (ibid., 117)
The early writers which attest to this apostolic view of the Eucharist need not be exhausted here, as the point is clear.
Therefore, in the New Covenant, the table/altar or table-as-an-altar (think of the homes used in the persecuted church for the Eucharist) on which is offered the Eucharist, has become the Table of the Lord ipso facto, by being the place where sacrifice is offered to God at fixed times. This is why St. Paul compares the table of the Lord with sacrifices offered to demons rather than to God. The sacrifices differ, and the gods differ, but not that it is a sacrifice—in which the parallel consists.
Jesus' Table in the Kingdom of Heaven
To be spiritualized are Jesus' references to a 'table' (Lk 22:30) and 'mansions' (Jn 14:2) and 'a marriage supper,' (Mt 22:2-14; Rev 19:6-9) in the Kingdom of Heaven. Both are in reference to the future when we are in heaven. They don't refer to St. Paul's concern about the Eucharist, which he relates in terms of the contrast with pagan sacrifices to their pagan gods.
"The Table of the Lord" in reference to sacrifices offered to Him (i.e. in contradistinction to altars whereupon are laid sacrifices to idols and false gods) is, make no mistake about it, a reference to the Altar where sacrifices are normatively offered to God.
He sees the Eucharist as a sacrifice to God is incompatible with ongoing idolatry and affiliation with paganism of any kind. Don't forget how groundbreaking and difficult the idea of abandoning idolatry in acceptance of the life of the Christian it is for these early communities, steeped in paganism!
"We Have an Altar"
Hebrews 13:10, like a few other references in the New Testament, seems to be purely spiritual, and yet might in fact be spiritualized references to actual physical realities.
Hebrews 10:13-16 (DRB)
10 We have an altar, whereof they have no power to eat who serve the tabernacle. 11 For the bodies of those beasts, whose blood is brought into the holies by the high priest for sin, are burned without the camp. 12 Wherefore Jesus also, that he might sanctify the people by his own blood, suffered without the gate. 13 Let us go forth therefore to him without the camp, bearing his reproach. 14 For we have not here a lasting city, but we seek one that is to come. 15 By him therefore let us offer the sacrifice of praise always to God, that is to say, the fruit of lips confessing to his name. 16 And do not forget to do good, and to impart; for by such sacrifices God's favour is obtained.
By Jesus we offer the sacrifice of praise. But it explicitly qualifies or explains what is meant by 'sacrifice of praise,' namely, the fruit of the lips confessing His name—prayer, worship, including the liturgical. If we take this to exclude the Eucharist's sacrificial nature, we make a mockery of the Old Testament and the New, wherein realities are spoken of symbolic and spiritual terms even when the realities themselves also exist. It would be strange for one apostle to make reference to eating the sacrifices of the altar in reference to the Eucharist, and then another use it in an exclusive sense not for the Eucharist—instead of one more directly and the other more spiritual.
For example, he clearly doesn't mean it exclusively, but spiritually, when he explains the offering of sacrifice both as offering Him praise, and, a bit later, as going good, and giving, again identifying these as the true sacrifices pleasing to God.
Note that while St. Paul references the Eucharist in terms of an altar, it need not be held that he was referring to even a plain table on which the Eucharist was then celebrated (which was most probably the case, given the universal use of altars when this became legal among other things), but to the notion of the sacrifice in general taking place being comparable to the Old Testament altar and those who partook thereof (1 Cor 10:18).