Well, you would think that one would be easy to answer, wouldn't you?
First off, the obvious ones, poison ivy (Toxicodendron radicans and T. rybegii) and poison oak (Toxicodendron pubescens and T. diversilobum). At first I thought about searching the web over for copyright free images, but decided against it. Both plants have an eastern and western version in the United States and since they are wild plants they are greatly affected by their growing conditions. So trying to identify them from a picture of an ideal specimen could lead to tragedy. The poison in both plants is so long lasting it survives even in the dead fall leaves and in the smoke if it is burned. So, needless to say (you would think) you should not make steam prints or dye with it. For a mind blowing story about people who decided to try it out anyway, check out India Flint's blog post about such things-YIKES! I have discovered here in Eastern Washington poison ivy seems to like to grow at the feet of Oregon Grape, why I wouldn't know, but it does. So, if I am picking up windfall leaves, I avoid raking anything out from under an Oregon Grape as Oregon Grape leaves don't work anyway and who knows what else might be lurking around under there. Last summer in Northern California I finally saw poison oak for the first time and it also likes to grow up, in and around other plants.
Fact is, every area of the country has poisonous plants both wild and in our gardens. It is really important that you know what they are. Knowing the scientific names helps, check out the ones above-"toxic" should give you a clue! Any name ending in Tinctoria or Tinctorium indicates a dye plant, but not necessarily that it is harmless.
If you are the type of person that breaks out into hives just riding the lawn mower, this may not be the hobby for you. Many plants that do make color also have active ingredients that can give them medicinal properties, so anything that can help you can hurt you. Tansy, for instance makes yellow and green dye. It has been used as a medicinal tea but should only be used under the supervision of a trained herbalist as if consumed in large quantities it can cause convulsions and psychotic effects and is poisonous to grazing livestock and can poison the milk production from those animals. Rhubarb leaves contain oxalic acid, used both as a mordant and a cleaning agent, but is lethal to consume. Things like oleander, foxglove, hemlock and anemones are so poisonous they, like poison ivy and oak, should be avoided altogether. It is possible that what one person has a bad reaction to, another will not. A friend of mine was clearing scotch broom (also a known dye plant) from her yard and suffered anaphylaxis, making it to the emergency room just in time. She told me this after I had a bunch boiling away in a bundle; no harm seemed to come to me from it, however I think I will be more careful in the future-especially since the result from the steamed leaves was underwhelming.
The best advice I can offer is that if you are not an expert at plant identification it might be advisable to go on guided nature walks or take classes if possible. Park services, county extension services, garden clubs and other outdoor activities groups offer such things and they can be really informative as well as fun. Having a local park ranger point out what poison ivy looks like in your area is much more memorable than a picture. That being said having an actual guide book for your area along on a collecting trip can help you avoid danger. There are also weed apps for phones, but if you are out far enough, there may be no signal in order to be able to use it. That, and not everything that is dangerous is considered a "weed". Old school is best in some cases.
Using common sense when processing plants is invaluable. Wearing gloves while working with things you are unfamiliar with, keeping a dedicated set of tools for dyeing and doing all your cooking outside are good for a start. Remember that in some cases, nobody ever thought anybody would be cooking a particular plant to begin with, so no one knows what might be in the fumes, just make the assumption there might be something poisonous in the vapor and don't breath it in. The mordants you use can have issues as well, sometimes more so than the plants; wearing gloves can protect you from a lot of strife. Even things that are food stuffs can be problematic, my "roommate for life" has requested that I never cook cabbage for over an hour in the house ever again-he was coughing his brains out. After having to deal with cabbage scented sheets, curtains, rugs and blankets, it is a request I have no trouble complying with!
The long and short of it is, don't get a false sense of security from the words "natural" and "organic" there are plenty of naturally occurring dangerous organic substances in the world, it is up to us to use common sense in order to avoid trouble.