Exceptions:
If you have burned wood with paint on it, particularly old lead paint.
Pretty much any recycled timber, even new stuff, as it has probably been treated with all sorts of heavy metals that you don't want in your tomatoes.

Ash isn't particularly carbon rich, your fireplace tends to burn carbon, but there are clearly traces. Much of the ash is mineral salts in the timber that were not evaporated, and amorphous silica granules. Those salts can be what the soil is lacking in some cases, but clearly one could over-do it.
(Arsenic, Fri 13 Jul, 2:73 and, Fri 4 Mar 2011, 23:40, Reply)