Updated in December 2016. I tried not to duplicate the @Alien Life Form answer, but will add more information.
Short answer :
In most cases, OpenVZ hosting will use OpenVZ 6 with an outdated kernel that is incompatible with Docker.
The docker is supported only with OpenVZ 7 (based on the 3.x kernel, see https://openvz.org/Docker_inside_CT_vz7 ) or with OpenVZ 6 with the kernel version 042stab105.4 or later (see https://openvz.org/Docker_inside_CT )
Long answer :
Docker requires two features of the Linux kernel to work: management groups and namespaces. Therefore, you need a kernel with these features.
With OpenVZ you do not control your core, only a hosting company. Most hosting companies will not cooperate and upgrade the kernel, so if the kernel turns out to be incompatible, you are usually out of luck.
OpenVZ 7 is based on 3.10 kernels that support Docker out of the box, but I have not seen OpenVZ 7 hosting yet - KVM is probably the most popular virtualization with the advent of new low-cost providers on the market.
OpenVZ 6 is more common with older vendors and is based on the 2.6 kernel, which is generally incompatible with Docker. However, OpenVZ kernels are not normal 2.6 kernels, but have several functions passed from 3.x kernels. So in fact, the kernel 042stab105.4 and later versions support the functions required by Docker in accordance with the official OpenVZ wiki (see @Alien Life Form answer).
The text below applies only to OpenVZ version 6.
If uname -a shows the kernel 042stab105.4 or later, you can use Docker with some settings to install the necessary special file systems.
If he is older and the company is ready to cooperate, they cannot install the main core, since it is incompatible with OpenVZ. They must install a special kernel with OpenVZ patches from https://openvz.org/Download/kernel . Preferably, the latest stable version (i.e. 042stab120.11 at the time of writing), but not less than 042stab105.4 . For comparison, all the OpenVZ hosts that I saw have something like 2.6.32-042stab075.2 , which is not only incompatible with Docker, but also vulnerable. Therefore, you can try to use the vulnerability argument to force support to upgrade :)
Another obvious, but not always applicable solution is to switch from OpenVZ to hosting with another virtualization technology such as Xen or KVM. However, it may happen that the main CentOS 6 core does not have the necessary functions, so CentOS 6 can only be compatible with Docker when launched with loose fixed kernels. So you can consider upgrading to CentOS 7.