The meaning of this verse is one of the most significant issues today as it applies to various hermeneutical systems. In covenant theology there is no difficulty on their part in associating Paul with the new covenant in Jeremiah 31 and their replacement theology suggests the church is the new Israel. In that system the promises made to Israel are said to be fulfilled in the Church. The debate among covenant theologians then becomes an issue to what degree are those promises, including the new covenant spirtualized.
Among progressive dispensationalists there is the notion that the Church partially fulfills the new covenant as part of their "already/not yet" system of theology. So in this view Paul is suggesting that ministers in the church are fulfilling or carrying out the aspects that are being fulfilled now, with the bulk being fulfilled in the future by a restored Israel. Gunn (see below) and others point out that a great deal of importance is placed on this verse as a pillar of their system of hermeneutics.
Traditional dispensationalists have had a much more varied and difficult time dealing with this verse. George Gunn lists there proposals that have been put forward by traditional dispensationalists:
Dispensational View : Participation - The church, by fulfilling the Great
Commission, does not partially fulfill the new covenant, but does participate
in some of the blessings of the new covenant.
Dispensational View: Two New Covenants – The church has its own “new
covenant” with God that is distinct and separate from Israel’s new covenant
of Jeremiah 31.
Dispensational View: No Relationship - The church is not directly related to the new covenant in any way. The church is related to the Mediator of the
new covenant and to the blood of that covenant, but is not a participant in
the covenant itself.
George Gunn, "2 Corinthians 3:6 and The Church’s Relationship to the New Covenant" Click here
One of the central issues in the debate is the function of the genitive in the phrase διακόνους καινῆς διαθήκης. According to Gunn, if the phrase is an objective genitive then the phrase would mean: "those who minister (or 'administer') the new covenant," and that "Paul would be referring to the new covenant as the content of his ministry." Also according to Gunn if the genitive is a "genitive of description" then it would be translated as "new covenant-like ministers" and it "provides a helpful description of the kind of ministry in which he was engaged."
Gunn's conclusion, based on seven very detailed exegetical issues, (see the link above), is the following:
Having examined various exegetical/hermeneutical issues, it is my studied opinion that Paul was not describing the content of his message, but rather the manner in which he conducted his ministry. Ultimately, the chief exegetical/hermeneutical issue questions whether the epression διακόνους καινῆς διαθήκης represents an objective genitive or a genitive of description. A consideration of the referent of ἡμᾶς, the context of the statement, the use/non-use of the article, the theological viewpoint of author and recipients and the way in which Paul refers to the OT lead, I believe, to the conclusion that Paul’s point was that his ministry is a “new-covenant-like-ministry,” not that he was administering the new covenant. Reference was to the style of his ministry, rather than to the doctrinal content of the new covenant. Thus, this verse does not support any kind of a realized eschatology, or church participation in the new covenant.
George Gunn, "2 Corinthians 3:6 and The Church’s Relationship to the New Covenant" Click here
I myself have gone through various "phases" when it comes to the issue of the new covenant. At first I supported the two new covenant view, having been influenced by Miles Stanford and the early writings of Lewis Sperry Chafer. Today I would put myself more in the camp of those who argue that the New Covenant belongs to Israel alone. There are strengths and weaknesses of very one of the views on the nature of the New Covenant as it applies to the Church.