Firsthand experience here
I dye my hair every 4-8 weeks to wash out the grey hairs. From my personal experience, dying ones hair does damage the hairs. I've done some research on this, and chemically what is going on is that the peroxide in the hair dyes that is the solvent for the coloring does damage to the grown and mature hair shafts that are being dyed. The damage is part and parcel of the dying process, in order for the dye to set within the hair shaft, it needs to perforate the hairs so to speak to give the dye a place within to rest and leave its much wanted coloring agents.
The peroxide effectively damages the shafts, tearing open the intertwined keratin structures. Thus, dying thin hairs can theoretically make them more susceptible to breakage.
However, damaging the mature and grown surface hairs with the effect of adding pigment to them is far different chemically than any supposed claim that they add to the DHT/miniturazation process that causes male pattern baldness. In effect, from my personal experience, I stand very firmly behind the claim that the effect of peroxide in making hair look more pigmented has no documented effect in terms of adding to the process under the skin level that DHT is playing to make the hair growth that one has more minaturized. One can shave his hair completely, the ultimate in damage to one's surface hair, and yet still the DHT under the skin will continue to reap its damage to the hairlines of susceptible males suffering male pattern baldness.
My conclusion has been that good nutrition and good occasional conditioning will keep even dyed hair in excellent shape. Despite the occational dying of my hair, my regime of minoxidil, spironolactone, and Nizoral still continue to give me excellent results and reduction in my balding areas despite it. Just my two cents.
And if hair dye were so counterproductive to the effects of minoxidil, then why would they include coupons for Just for Men hair dye inside the packaging for Rogaine? If Just for Men hair dye counteracted the effects of Rogaine in any way, one would think that they would not only NOT include such coupons, but advise against the use of hair dye to enhance the supposed efficacy of their drug, Minoxidil, so they ensure more repeat business.