Is the hydrogenation of carbon dioxide to methane a solution or a problem?
The problem is that the carbon dioxide which is released from burning is often mixed with other things which include air. If we were to want to convert carbon dioxide into methane then the oxygen of the air would make it harder.

While one option might be chemical looping combustion to obtain a purer carbon dioxide stream we still have the problem of where would we get hydrogen from. The normal method in industry to get the hydrogen gas needed for things like hydrogen peroxide production is to use steam reforming of methane.

$\ce{H2O + CH4 -&gt; CO + 3H2}$

Using the water gas shift reaction the carbon monoxide is converted into carbon dioxide and hydrogen (using more steam). My worry about an industrial process for the conversion of unwanted carbon dioxide into methane is that unless we use hydropower, solar power or nuclear power to obtain electricity for the production of hydrogen from water then we will never be able to make a net reduction in the amount of carbon dioxide as the process will form as much carbon dioxide as it consumes.

There is some electrochemistry in which carbon dioxide can be converted into other things by reduction at the cathode but I think that this is currently only at a very small scale.