The short comment is: please don't do this to your Akhal Teke!
However, you are presumably here for the longer explanation...
The probably worst singular thing which happened to the breed since Turkmenia fell in 1881 were Sergey Filatov's repeated Olympic and Grand Prix dressage wins on sire 668 Absent. They created the myth of the "dressage prospect Akhal Teke" and ensured the subsequent breeding and selection for warmbloody looking Akhal Teke with large calibre, high-stepping gaits and great height in a Soviet effort of cashing in on his successess.
There are a few things that the general interested public does not know about Absent and Filatov. Absent was not at all an easy ride. Several capable dressage specialists tried to work with him and failed miserably, the stallion fought them, sulked and created repeated havoc in the dressage arena. Filatov managed to appease Absent's rebellion against repetitive dressage training by instituting a high rate of diversity, rarely executing the same lesson twice during a session and refining movements only with long (up to several weeks!) intervals between them. It takes an equestrian genius like Filatov to pull such a stunt off at the kind of level Absent competed in. Also, Absent himself was an impure Akhal Teke with several ancestral lines going back to English sires in his back pedigree via his dam Bakkara. Given a close look at him and his pure sire Arab, it becomes quite clear where those English lines had most influence.
Today, it can be considered highly doubtful that Absent would manage to even make the entrance qualifications for such events. Classic sport competitions, especially those of dressage, have largely changed between 1960 and now. The warmbloods have become the performance benchmark regarding conformation, gaits and comport to such a degree, that even the erstwhile mainstay of dressage, the baroque breeds like Lippizzan, Andalusian and Lusitano, have a very hard time keeping up with them. Special breeds are extremely rarely found in dressage past entrance level competitions, even rarer are their successess. At International Grand Prix level the prizes go to warmbloods, to be more precise, mainly to Dutch and German warmbloods. A look at the pony classes of that level will show you that here too smaller versions of warmbloods are employed where the money is made.
While, theoretically, even a Shetland pony or a Shire Horse ought to be capable of demonstrating the required lessons, few people would dream of trying to compete at high levels with them, though there indeed are excellent riders who have trained individuals of both breeds to Grand Prix level. The general, serious riding public regards this at best as astonishing application of the rider, at worst as circus stunts. To a lesser degree this can be said to be the case for other special breeds like Arabians, Akhal Teke or Quarterhorses as well.
Of course the Akhal Teke can be ridden and trained in dressage, it is a horse which will do literally anything for its rider, when wisely managed and well-bonded. And also of course a basic training in dressage, as in teaching contact, collection, impulsion, rhythm and relaxation, whether in the dressage arena or out in the open country, whether classic or western dressage, is good and wise basic training for any horse. As long as the Akhal Teke is allowed to engage in his natural gaits and may position himself as his conformation allows for, there is no problem with such basic training.
It is when the aim is competitive classic dressage and winning such competitions, that perversion sets in. We are here dealing with a horse bred for the open country, a horse which was selected over several centuries, if not millennia, to independantly decide how it covers the distance between A and B, at a fast speed, a horse which has been bred to scan the horizon with its far sight and a horse which, when typical of pre-1900 times, has conformation quirks which cause every dedicated dressage rider and breeder of modern dressage horses to shudder. Its most original gaits are flat and even, it is built for the canter, not the trot, up to 1900 many Akhal Teke even were exquisitely gaited horses with a good running walk.
While an Akhal Teke will submit to dressage training, because it will do anything for its rider, that does not mean, that this work suits it. How happy is a horse bred to speed across the plains, intelligently working for its rider's best, alert for attack from animal or man, consistently staring at the next 3 yards of dirt ahead, while executing repetitive movements within a space of 40 by 20 metres? The vast majority of Akhal Teke shown in dressage do not "shine", there is no inner glow of contentment, expression and satisfaction, something easily and readily found in just about any Lippizzan or Andalusian in the arena, horses which - contrary to the Akhal Teke - all but scramble to present their riders with perfect dressage movements and lessons, the moment they sit deeply into their saddles.
What is even worse in this respect is that all this is for nothing. Past a set level no Akhal Teke today is capable of beating the warmbloods, the ranking list of the FEI for dressage shows exactly one Akhal Teke at that level, and it is nearly last, well below the baroque horses, which inspite of their genuine aptitude for dressage also are beaten by the warmbloods and not just by a margin either.
Lastly, and most seriously, it is this misbegotten and misdirected idea of engaging in dressage competitions, which has caused and still causes a serious deviation in breeding selection and grading of the Akhal Teke, contributing to further alienating the breed from its original Turkoman status and purpose.
What is very difficult to grasp is the motivation behind trying to compete Teke in dressage. They are not going to achieve there, if original in character and type they positively hate the most common training methods, they never give the air of enjoying themselves while executing the lessons, and even though they may be comfortable to their riders, that is no reason to twist them once past their ABCs.
Anyone wanting a good dressage prospect can go ahead and buy one, for a lot less money than what an Akhal Teke costs. Good young warmbloods are available as low as 3000 U$ from smaller breeders in Europe, confirmed prospects, broken to the saddle and already showing their potential, still are only 10,000 to 12,000 U$ these days. A fancy dressage horse? Go have a look at the lighter Andalusians or Hispanos. And if someone insists on the Russian touch, there are few Russian breeds better suited to dressage than the Tersk horse, which comes along with the looks of a more agile and finer-grained Siqlawi Lippizzan, complete with a silver or golden metallic sheen on their coats, lots of endurance, and endowed with phenomenal gaits, great natural collection and a definite penchant for such work, these horses also are much less pricewise than an Akhal Teke.
In the end, once again: spare them that kind of work, let them excel where they can excel and have fun doing so!