"According to hair biology experts and styling experts alike, grey hair is more resistant to color than younger hair because of its texture. The relative lack of natural oils in the hair compared to younger hair make it a rougher surface that tends to reject the color being applied, especially around the roots."
I've been dying my hair with semi-permanent color for years, and it does *not* take on the grey/white hair. When I dye my hair red, the silver hair just turns pinkish and then it washes out in days rather than weeks. I already have very dry hair though, and the colorless hairs are naturally even drier, so ymmv. The healthier your hair is, the better it will hold on to dye.