Just to clarify, this is the cooperative threat reduction program, otherwise known, more popularly, as the Nunn–Lugar initiative. This is an initiative that, as you say, has been going on for 20 years, whereby the Americans have been providing capital to decommission many of the excess Russian nuclear warheads, and indeed take a lot of the fissile material and burn it in reactors in the United States to produce electricity: atoms for peace, you could say.
My observation is that this is a symptom of the declining trust that the Russians have of the whole process. They're spiking their noses, frankly, on this, because the types of warheads that are being decommissioned are very outdated and far from being relevant to any kind of Russian security, whatever one's opinion is of what's good for Russian security.
I think it's a political football rather than a strong objection within Russia to the program itself. I'm not aware of those objections being present other than the fact that it is part of the indication that the Americans won the Cold War, and it's symbolic, therefore, within Russia. This withdrawal from the program I perceive to be a negative symbol of the potential for Russian cooperation, and it doesn't bode well for the future.