Reading some of Tim Keller's books recently, certain emphases stood out. One being something he calls the "divine dance." Keller prefers framing discussions of the Trinity with this analogy. He concludes with it in The Reason for God and opens with it in King's Cross (aka Jesus the King: Understanding the Life and Death of the Son of God)...
Exploring the background to this analogy, I came across a very helpful book, Engaging with Keller: Thinking Through the Theology of an Influential Evangelical, contributed by various authors, including Pastor Kevin J. Bidwell.
In chapter three, Pastor Bidwell delves into Keller's "divine dance," highlighting significant problems with the analogy. Engaging with Keller has been reviewed several times and I've noticed this chapter three receives the brunt of reviewers' criticism. Reading those criticisms, though, the criticisms seem to be due to our ignorance of the doctrine of the Trinity and our sloppy thinking.
Pastor Bidwell studied under Robert Letham for his M.Th. and Ph.D. Letham is one of the most helpful contemporary theologians on the doctrine of the Trinity with a knowledge of historical theology that is vast, deep, and very helpful to the church. And from these two students of the doctrine of the Trinity, we learn that Pastor Keller's divine dance isn't simply a weak analogy. Rather, it is Keller's paradigm for the relationship within the Trinity, as well as man's relation to the Trinity in redemption.
As Pastor Keller states it, the "dance" of the Trinity precludes any notion of self-centeredness. Rather, it is a constant voluntary deferral from one Person to the other Members of the Trinity:
The life of the Trinity is characterized not by self-centeredness but by mutually self-giving love. When we delight and serve someone else, we enter into a dynamic orbit around him or her, we center on the interests and desires of the other. That creates a dance, particularly if there are three persons, each of whom moves around the other two. So it is, the Bible tells us. Each of the divine persons centers upon the others. None demands that the others revolve around him. Each voluntarily circles the other two, pouring love, delight, and adoration into them. Each person of the Trinity loves, adores, defers to, and rejoices in the others. That creates a dynamic pulsating dance of joy and love. The early leaders of the Greek church had a word for this—perichoresis. Notice the root of our word ‘choreography’ is within it. It means literally to "dance or flow around." 
Although Pastor Keller references both C.S. Lewis and Cornelius Plantinga in support of the particularities of his "divine dance," he's hanging his hat on Plantinga. Pastor Bidwell writes:
Such a concept (of mutual deferral) is not to be found in the Nicene Creed or the Reformed confessions and catechisms...One place such language is to be found, however, is in the "Men, Women, and Biblical Equality" statement of which Plantinga is listed as an endorsing signatory. The statement includes the following: 'In the church, spiritual gifts of women and men are to be recognized...in pastoral care, teaching, preaching, and worship [...] In the Christian home, husband and wife are to defer to each other ..." 
Pastor Bidwell goes on to explain that Plantinga is simply projecting his egalitarianism onto the Trinity.
"Ah", you say, "but that's only Plantinga, Keller is a complementarian."
Sure, Pastor Keller is a complementarian...
"Perhaps Pastor Keller is simply rehabilitating a helpful concept. Perhaps he's appropriating helpful ways of communicating timeless truths from contemporary sources," you say?
But then again, maybe Keller is accommodating God's truth to the preferences and tastes of his contemporary audience. No one of a metropolitan persuasion would find a divine dance offensive. It would reaffirm their sensibilities.
Then too, Keller's definition of 'perichoresis' is wrong. Perichoresis does not "literally" mean "to dance or flow around." Neither in etymology nor historical theology is there any basis for a connection between perichoresis and choreography. Rather, this etymological error seems to have been the brainchild of Feminist theologians such as Catherine Mowry LaCugna.
Pastor Bidwell points out that LaCugna admits there's no philological warrant connecting dance to perichoresis, but she links them together anyway. Why?
The divine dance imagery eliminates the divine ordering of the three Persons of the Trinity.
Some might defend Keller by saying he's simply highlighting the dynamic and perpetual activity of the Trinity over and against a static view of God. Perhaps, but if it's all about dynanism versus stasis, Pastor Bidwell highlights how Keller's "divine dance" portrays the wrong kind of motion:
There is a dynamic movement between the persons of the Trinity. This is the act of the Father eternally begetting the Son, and then sending him as the God-man, along with the action of the procession of the Holy Spirit. These movements do not portray the being of God as static, but that of one who is "outward-moving." 
This is the Trinitarian motion ordered according to God's Word.
Living deep within egalitarianism, though, it should surprise no one that Keller's "divine dance" hamstrings the Trinitarian motion revealed by Scripture. Bidwell points out the crucial question is "How are we to distinguish the three persons..."
Keller's "divine dance" of perpetual deferral does not describe the Trinitarian God revealed in the Bible.
In Keller's "dance," it's impossible to distinguish "who's who" among the three Persons of the Trinity.  His dance eliminates ordering, leaving each Person of the Trinity interchangeable with the other Members of the Trinity. Bidwell notes:
The notion that the Trinitarian persons 'defer to one another' is inadequate to handle the teaching that Christ is sent by the Father, and that the Son, as mediator, obeys the Father, he does not 'defer to' the Father. 
Pastor Bidwell raises many other problems with Keller's construct, doing so with clarity, helpfulness, thoroughness and great organization—all in approximately 30 pages, which is no small feat. I commend this book to you.
There would be very little reason to criticize Tim Keller's use of the dance analogy if he used it in passing to communicate a singular truth. Instead of an aside, though, this is Keller's paradigm for the Trinity. And within Reformed circles, Keller's egalitarian paradigms are très chic.
 Reason for God, pp. 214-215
 Engaging with Keller, pp. 110
 ibid, pp. 114-115
 ibid, pp. 117
 ibid, pp. 119: "Without this ordering of the persons, all you have are three interchangeable persons having names that mean nothing. The 'divine dance' teaching that lacks the doctrines of the eternal generation of the Son and the distinguishing relational properties of the persons of the Trinity thereby introduces theological weakness into the doctrine of the Trinity, with implications for Christology. This metaphor then, does not serve to enhance our appreciation for the doctrine of the Trinity, it undermines it."
I have noted elsewhere this same tendency in Reformed circles. I would have liked for Pastor Bidwell to delve into the Christological implications, but that likely exceeded the purview of his chapter. One possibility may include an implicit Nestorianism when shifting the metaphor of dance with regard to the Son to His incarnation. Another possibility may be a form of Adoptionism where the appellation of "Son" is in consequence to His incarnation (both could actually be implied).
 ibid, pp. 123