Like almost all the do not feed items, the risk is low as long as you give in moderation... A tomato plant here or there isn't going to cause any harm, feeding your chickens tomato plants as a sole diet for a week most certainly might or feeding it to them regularly might...
If you actually Google up the toxicity of tomato leaves/stems you will see a lot of the claims are based on urban myth because it's related to the nightshade family... That isn't to say it doesn't contain some toxins (as most things do) but it's not nearly the demon toxin many make it out to be...
http://www.thekitchn.com/are-tomato-leaves-actually-poisonous-222259