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:
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.
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
All Scripture is quoted from the King James translation.
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:
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.