As God’s co-workers we urge you not to receive God’s grace in vain (2 Corinthians 6:1).

How is it possible for anyone to receive God's saving grace in vain? To find the answer to this, we should look to see if Paul provides any clues as to what prompted this urging. In 2 Corinthians 5:12 Paul writes,

We are not trying to commend ourselves to you again, but are giving you an opportunity to take pride in us, so that you can answer those who take pride in what is seen rather than in what is in the heart.

There appears to be some wrongful influence in the church that the Corinthian believers are called to contend with. Later on Paul would be more explicit:

But I am afraid that just as Eve was deceived by the serpent’s cunning, your minds may somehow be led astray from your sincere and pure devotion to Christ. For if someone comes to you and preaches a Jesus other than the Jesus we preached, or if you receive a different spirit from the Spirit you received, or a different gospel from the one you accepted, you put up with it easily enough. I do not think I am in the least inferior to those “super-apostles"... And I will keep on doing what I am doing in order to cut the ground from under those who want an opportunity to be considered equal with us in the things they boast about. For such people are false apostles, deceitful workers, masquerading as apostles of Christ (2 Corinthians 11:3-5, 12-13).

In light of this Scripture, the problem facing the Corinthians was as serious as compromising the true Gospel, the true Spirit, and the true Christ. If the Corinthians were to "be led astray" by these false teachings, they would destroy their witness for Christ. 

In 2 Corintihans 5:19 Paul writes,

God was reconciling the world to himself in Christ, not counting people’s sins against them. And he has committed to us the message of reconciliation.

It is this commission and message of reconciliation that concern Paul. Paul writes that he and the Corinthian believers have been given the commission to proclaim the message of reconciliation. I have taken some liberties in translation by adding a colon and quotations to  suggest that perhaps the words beginning with "We implore..." and ending with "righteousness of God" could be the very appeal that Paul writes that God was making through them:

We are therefore Christ’s ambassadors,as though God were making his appeal through us:

"We implore you on Christ’s behalf: Be reconciled to God. God made him who had no sin to be sin for us, so that in him we might become the righteousness of God" (2 Corinthians 5:20-21).

Then, Paul's appeal in 2 Corinthians 6:1, namely, "As God’s co-workers we urge you not to receive God’s grace in vain," would be an appeal to the Corinthian believers with respect to their commission to preach the message of reconciliation. So I take the word "grace" not to mean saving grace but rather serving grace, that is, the grace given as a calling to ministry. For example, Paul writes elsewhere,

I became a servant of this gospel by the gift of God’s grace given me through the working of his power. Although I am less than the least of all the Lord’s people, this grace was given me: to preach to the Gentiles the boundless riches of Christ (Ephesians 3:7-8).

In other words, he tells them not to receive this commission, their calling to the ministry of reconciliation, in vain. In what sense would it be in vain? If they preached a different Jesus, a different Spirit, or a different Gospel, then it would be in vain. What is more, and related to this, if they failed to recognize that what the prophets foretold with respect to the time of God's favor and salvation was in their time, their ministry would be vain:

For he says, “In the time of my favor I heard you, and in the day of salvation I helped you.” I tell you, now is the time of God’s favor, now is the day of salvation (2 Corinthians 6:1-2; citing Isaiah 49:8).

This hearkens back to what Paul wrote in 2 Corinthians 5:16-18,

So from now on we regard no one from a worldly point of view. Though we once regarded Christ in this way, we do so no longer.  Therefore, if anyone is in Christ, the new creation has come: The old has gone, the new is here! All this is from God, who reconciled us to himself through Christ and gave us the ministry of reconciliation

Notice the temporal markers (i.e. "from now on," "no longer," "has come", "has gone...is here"). This relates to "now is the time" and "now is the day" in 2 Corinthians 6:2 and is a reference to the present ministry of reconciliation to which the Corinthian believers have been called to do. 

So, what did Paul mean by urging the Corinthians not to receive God's grace in vain? He meant for them not to forsake their present calling to preach the message of God's recociliation. He also meant for them not to corrupt that message by listening to those that would preach a different Christ, a different Spirit, or a different Gospel from what they heard from Paul.