I am by no means an authority on matters Orthodox or Tai chi, but I will tell you this...
I practice the kung fu style of Wing Chun (Ng San Lung Fa for those who are interested). So far I have learned three forms being Sil Lum Tao, Sil Lum Fut Ga, and Chum Kiu. These forms teach you the different techniques used in Wing Chun but also focus your qi (or chi). The movements, when slowed down are very much like Tai Chi movements and postures. Focusing ones chi gives themselves better self-control and really helps to relax them, in fact there have been some times after doing three full set of Sil Lum Tao that I have almost fallen asleep from being so relaxed. Any way, my point is that you can do Tai Chi by doing any kung fu style that has forms to focus qi.
I am lucky in that the school I attend keeps a Christian focus (one teacher always opens and closes with prayer). The qi is ones internal energy, like the energy you have through out the day. Doing these forms can help you strengthen you qi (read give you more energy), and keep you healthy (probably by the breathing techniques oxygenating your blood) . Personally I do not see a problem doing something that focuses on your body's energy, as opposed to something like, say, reiki that is supposed to be the practitioner channelling the "universal" energy that is in and around everything.
One thing I like to do is a set of three of any of the forms (usually Sil Lum Tao or Chum Kiu) before my prayers to help relax me and put me into a meditative state, this makes it extremely easy to focus on my prayers. One thing that I have recently read (through the links posted above) is that some people say the Jesus Prayer when doing Tai Chi and that is something that I am going to have to incorporate during my forms. Again, personally I see no problem doing a Tai Chi form and being an Orthodox Christian, but I am not a priest.
I'm glad someone responded, now I can make my joke:
Q. What is the EOC stance on Tai Chi?
A. Crane Style.
I hope the OP sees my humour as innocent.