I've heard that the inks used on newsprint are generally soy-based and thus suitable for use in vermicomposting bedding. Presumably the ink used to mark corrugated cardboard boxes is also safe, since this is another commonly recommended bedding substance. The inks used on glossy newspaper inserts, on the other hand, are said to be metallic and thus hazardous. What about other inks, for example permanent markers?
I would definitely avoid introducing these inks into the food chain; I think my answer to Cereal boxes for mulch? is relevant here:
Cereal boxes have been in the news recently, in Europe. According to the latest research, they may leak harmful mineral oils into the foods they contain - oils which originate mainly from the ink in the recycled newspapers that are used to make them and, to a lesser extent, from the printing ink on the box, and may be carcinogenic.
As yet, there is no hard evidence that these mineral oil hydrocarbons that migrate into our foodstuffs are a health hazard ; and even if they are, those that are present in the cardboard might well be converted into harmless substances on contact with the soil, over time. However, given the potential risk, I would definitely avoid introducing them into the food chain by using cereal boxes either as a mulch or a composting material.
There is a very full article about the issue by the UK National Health Service here.