How fast is the Cosmic Microwave Background Radiation (CMBR) changing?

This is a great question, and it also came up at a recent Planck conference: according to this blog post (see the paragraphs above the CMB gif), simulations suggest that minute changes in the CMB might just be detectable in as little as 100 years from now.

Just on dimensional grounds I would expect that $$\text{angular size of fluctuation in radians} \times \text{time since recombination}$$ is the time scale for significant changes in the CMB (where you choose what makes a fluctuation "significant" and select the angular size on that basis).

Why? Because the size of the regions defined by the fluctuations is the distance to the observed shell, and that distance is given by the time since recombination and the speed of light; and information can travel across those regions no faster than light.

The whole spectrum will cool a bit faster than that, but the cooling can be expected to be uniform. The rate of cooling is given by the Hubble (non-)Constant.

Note that the time since recombination is roughly 13.5 billion years, so even though the domains are pretty small it is still going to be a long wait.