If ash from charcoal briquets, a not very good idea :-) These are permeated with petrochemicals other than just charcoal to hold them together and maintain shape and to encourage ignition and even burning. A good, hot composting operation would very likely neutralize any ill effects but why take the risk?
There are all kinds of chemicals in "cardboard" but some seem to be in love with it.
Any problems with the ash are problems that come with any ash.
RpR, what "chemicals" do you think are in cardboard? Cardboard is simply paper, made from unbleached wood pulp. The adhesives used to hold it together are most often approved for use on food packaging meaning they are safe for human consumption.
Whether to use the ash from a charcoal grill in your compost or not depends to a great extent on what you used to grill. The molded briquettes have many things other than wood in them that you may not want in your garden, binders to hold them together, stuff to aid burning, etc. The chunk charcoal does not have those things. Both, however, will have grease from the foods grilled, fats. Like all wood ash this ash will be quite alkaline and that alone could cause problems with the bacteria that are digesting the material in your compost.
Most every one that knows about composting that I know think adding ash from your backyard grill to your compost is not a good idea.
If you think that cardboard, the real stuff or corrugated, is made by some magic with "organic" components only, welcome to Never Land.
I have burned some cardboard, and have noticed that some is now rather difficult to ignite.
There is a reason and it is not magic.
I would bet chemicals have been applied to make it less combustible, for safeties sake.
Some cardboard is treated with flame retardants, although those should not be readily available to the public. A flame retardant will make it difficult to lite that cardboard and they are "chemicals" no one should want in their soil.
During the manufcturing process of cardboard nothing is added that would be a problem in your garden. Cardboard used to ship food would not be so treated.
Since a lot of cardboard is made from recycled materials though, the overall makeup can't be totally inert. Like all large scale paper manufacturing processes, chemicals are used. If available, just recycle it.
Getting back to the ash question, I would generally advise against using more than small amounts of BBQ ash in the compost or garden. The aforementioned alkalinity is one reason, and another is the use of borax as a binding agent. Excessive boron in soil can affect plant growth.
As for the 'petrochemicals' including lighter fluid and the paraffin wax used in self-lighting charcoal, it's pretty much burned off during use so it's not much of a concern.
This comes from the Kingsford web site,
"Can you use this product as compost or fertilizer?
No. Both KingsfordÃ¯Â¿Â½ and KingsfordÃ¯Â¿Â½ Match LightÃ¯Â¿Â½ briquets contain ingredients other than charcoal to make them efficient cooking fuels. Charcoal briquets do not aid in the breakdown of organic matter."
Odd choice of words there, about not aiding in the breakdown of organic matter. Of course it's true, but there are so many other things they could have said that seem more...relevant.
I burn 7 or 8 of the 20 lb 18 lb 17 lb 16 lb bags of Kingsford charcoal every summer, there on the back concrete patio in the shade of the big ash tree, using an hibachi and one of those smoker thingies. I take a garden hose and rinse out the ash the next day, onto the grass, right at the edge of the patio. I've done this for, oh, 15 years now, and the grass shows no signs of disease or anything.
One year I tried using their mesquite charcoal. Did the same thing, and the grass yellowed up for a few weeks.
IOW, there is charcoal, and then there is other charcoal.
I have no explanation for that. It may be that the formulation is different, resulting in some difference in the ash. My impression is that cheap charcoal has more coal in it, but is the mesquite stuff 100% mesquite or are they just replacing the wood (usually oak) charcoal portion of it with mesquite charcoal? Either way, pH wise, ash is ash, pretty much. Interesting effect you observed there.