What feast is Paul referring to?
Therefore let us keep the feast, not with old leaven, nor with the leaven of malice and wickedness, but with the unleavened bread of sincerity and truth. (1 Corinthians 5:8 NKJV)
The preceding verses answers your question:
6 Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump?
7Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us:
8Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.
Verses 6 and 7 are referring to the Passover Feast. During Passover, Jews at the time (and today) would eat unleavened bread, as a symbol of moral purity. Here Paul is explaining this symbolism. In the analogy, leaven (yeast) represents sin, and he says that just as even a little yeast is sufficient to leaven a large loaf of bread, just a little sin (or sexual impurity in this specific example) is enough to contaminate an entire body of believers.
The feast referred to here is the Lord's Table or Communion. Paul would not be telling the Gentile Corinthian believers to practice the Old Covenant Feast of Passover or any other Old Covenant feast.
One answer here is mostly correct. They point to the fact that the feast is Christ. But that misses the point here. The point is that our communion with Christ is a made real for us in the New Covenant version of the Old Covenant Passover which is Communion and that we should not corrupt that by allowing people in open rebellion to participate. The close context here is that of ex-communicating a sinning brother from the fellowship. This would mean he would be bared from the Table and so the admonition here is that the communion feast be kept in a pure way, without the taint of sin.
This very passage states that:
For indeed Christ, our Passover, was sacrificed for us. 1 Cor 5:7b
We also know that the first communion supper in the upper room was instituted during the Passover feast. The passover feast which is somewhat directly referred to here is not the Old Covenant one but again the New Covenant Table of the Lord.
It would appear that the feast Paul is referring to is having Christ, look back at:
1st Corinthians 4:5 through 13 King James translation
Therefore judge nothing before the time, until the Lord come, who both will bring to light the hidden things of darkness, and will make manifest the counsels of the hearts: and then shall every man have praise of God. And these things, brethren, I have in a figure transferred to myself and to Apollos for your sakes; that ye might learn in us not to think of men above that which is written, that no one of you be puffed up for one against another. For who maketh thee to differ from another? and what hast thou that thou didst not receive? now if thou didst receive it, why dost thou glory, as if thou hadst not received it? Now ye are full, now ye are rich, ye have reigned as kings without us: and I would to God ye did reign, that we also might reign with you. For I think that God hath set forth us the apostles last, as it were appointed to death: for we are made a spectacle unto the world, and to angels, and to men. We are fools for Christ's sake, but ye are wise in Christ; we are weak, but ye are strong; ye are honourable, but we are despised. Even unto this present hour we both hunger, and thirst, and are naked, and are buffeted, and have no certain dwellingplace; And labour, working with our own hands: being reviled, we bless; being persecuted, we suffer it: Being defamed, we intreat: we are made as the filth of the world, and are the offscouring of all things unto this day.
All Scripture is quoted from the King James translation.
John 6:48 I am that bread of life.
Luke 4:4 And Jesus answered him, saying, It is written, That man shall not live by bread alone, but by every word of God.
Luke 9:3 And he said unto them, Take nothing for your journey, neither staves, nor scrip, neither bread, neither money; neither have two coats apiece.
Luke 11:3 Give us day by day our daily bread.
John 6:31 Our fathers did eat manna in the desert; as it is written, He gave them bread from heaven to eat.
John 6:32 Then Jesus said unto them, Verily, verily, I say unto you, Moses gave you not that bread from heaven; but my Father giveth you the true bread from heaven.
John 6:33 For the bread of God is he which cometh down from heaven, and giveth life unto the world.
Paul is indeed referring to the Passover, but in particular that part of the feast of which many Christians are unaware: the requirement to consume no leaven in the seven days following the Passover meal.
14 So this day shall be to you a memorial; and you shall keep it as a feast to the LORD throughout your generations. You shall keep it as a feast by an everlasting ordinance. 15 Seven days you shall eat unleavened bread. On the first day you shall remove leaven from your houses. For whoever eats leavened bread from the first day until the seventh day, that person shall be cut off from Israel.
If the feasts are types of the redemption of Christ, which Paul explicitly asserts in 1 Corinthians 5:8, the unleavened bread observance of the seven subsequent days may be a type he was drawing of the Christian life, meant to be lived out in an unleavened way, so to speak. We should remember that for Paul, and even for many Jews to this day, to "keep the feast" would not merely have meant consuming the Lamb, but ridding one's home of leaven and refusing to consume it for that following week.