Certain weed roots should never be put in the compost especially bindweed. Many years ago I had a garden with an infestation of Japanese Knotweed, before it was compulsory to report it. I successfully disposed of it by cutting it back to a few inches and filling the hollow stems with rock salt, it never came back but nothing grew in that corner of the garden for a few years. Getting back on topic, carbon monoxide is the product of incomplete combustion. If you have to remove a large ammount of ash the wood burner isn't burning properly and producing CO. There can be several reasons for this but most often its down to the flue not being swept often enough.