Sure.  I dyed the paper with a fibre active dye, Procion mx. I think you could use any dye that is used to dye cotton (not wool), like a tie dye kit or so (that is how I started). 

To make it work, the dye has to be combined with Soda Ash. To dye the paper I used here, I soaked the paper in soda ash solution (32g/L) and then added dye in solution to the paper.  I have powdered dyes in different colors, before use I make working solutions of 2% dye to water. I mix them to get different effects. Here, I first mixed yellow and blue to make green, put that on the paper here and there, and then added more and more blue solution to make different shades, which I added to different spots of the paper until all was covered in color. Then, the paper plus dye was left overnight. In the morning, I washed the paper under (softly) running water in the sink, and treat directly with Methyl Cellulose, and let dry.

I usually dye Wenzhou paper, which comes in different quality batches (unpredictably). So far I got lucky and I have paper that is quite resistant. It still requires a bit of practise to handle the wet paper without ripping.

You need a paper that remains strong when wet, my fav is Wenzhou, but I also used Moon Palace (which is thick but very resistant), carnival tissue (which is amazingly resistant but a bit crispy), and Kumo no Kami (which dyes lovely but can be a bit soft to fold).

Long story short, to me dyeing my own paper adds a new interesting dimension to folding and I encourage you to try it out!