There's not an issue with painting composite structure, it's the colour you paint it which is important. And when you're talking about issues with the "composite" more specifically you're talking about the matrix (resin) part of the composite.
The matrix is susceptible to degradation from a number of environmental sources, particularly IR, UV and condensation. These can cause degradation of the matrix structure and lead to a reduction in the physical properties of the matrix. IR is the most obvious of these (i.e. if you put your hand on a dark part of a wing and a light part you'd be able to feel the temperature difference). UV is less noticeable but in the long term equally as damaging.
Seeing as the matrix provides almost all of the composite structures compressive strength, over time exposure to excessive IR and UV radiation could lead to a reduction in static strength to the point where a manoeuvre (e.g. a loop), which was previously well within the material’s capability, causes failure of the material.
So yes, you can paint composite structures whatever colour you like but you either need to take property degradation into account or use some form of inhibitor (for UV in particular).