How much radiation would something need to output to sufficiently affect the CMB?
It is not very clear what you mean. There are many cosmic sources of microwave radiation that emit so much energy and/or are so close to us that they confuse measurements of the cosmic microwave background. For example "compact sources" such as giant molecular clouds, supernova remnants, H II regions; "diffuse" sources due to dust in the galaxy and in our solar system. In order to get a clean signal, these foreground sources have to be removed from the data. There are also more subtle cosmological effects caused by lensing and gravitational time dilation caused by the large scale structure in the universe.

There are many ways and techniques that are used to characterise and remove these backgrounds (e.g. see Ade et al. 2013).

The most confusing background source would be one that is diffuse, spread over the whole sky almost uniformly and with a spectrum that is very similar to the cosmic microwave background. The source that most closely matches these properties is dust in our own solar system. An imperfect subtraction of this may have led to controversy over possible large scale anisotropies in the CMB- for example see Schwarz et al. (2015); who discuss the problems of removing the signatures of solar system dust.