Pigments can affect mechanical properties of plastic parts. Organic pigments have been shown to act as nucleating agents. Different colour pigments are due to different chemical structures and therefore different colours have different effects on plastic part properties. Phthalocyanine pigments (blue and green) in particular would affect mechanical performance. There is plenty published literature on pigmented injection moulded parts to give you a broad overview of the issue.
I would not think different pigment chemistry would influence energy consumption. However, different pigments require different masterbatch formulations and loading. Therefore influence of polymer carriers may be on the processing temperature I.e. polymer blend effects.
As I worked with different FDM printers and materials I can help you with my experiences but there isn't any scientific basic back of these experiences. all in all color plastic filament because of higher impurity needs higher temperature and normally lower adhesion between layers.
I suggest that printing with raw material and at the end post processing and painting result better.
Recently reviewing FDM process parameters influence over specimens' mechanical properties, I found the following scientific paper which can answer your question:
1. S.-H. Ahn, M. Montero, D. Odell, S. Roundy, P.K. Wright, Anisotropic material properties of fused deposition modeling ABS. Rapid Prototying Journal, 8(4) (2002) 248–257.
2. B. Wittbrodt, J.M. Pearce, The effects of PLA color on material properties of 3-D printed components. Additive Manufacturing, 8 (2015) 110–116.
3. N.G. Tanikella, B. Wittbrodt, J.M. Pearce, Tensile Strength of Commercial Polymer Materials for Fused Filament Fabrication 3D Printing. Additive Manufacturing, 17 (2017) 40-47.
in my paper found on : Article Strength to cost ratio analysis of FDM Nylon 12 3D Printed Parts
you will find a table summarizing and ordering the most significant parameters studied(around 17 parameter) during the last 16 years (represented by selected 15 articles), The table also summarize which responses (Mechanical, Geometrical, and Cost)were studied in each article and the material used.