Tomatoes: the fruits will not hurt chickens. So if you get one (or a bunch) that split open or have bad spots, give those to the chickens.
Potatoes: the green plants should not be given to chickens.
Cooked potatoes (peels and insides) are fine for chickens.
I've always thrown potato peels into the "compost pile" that is the chicken pen, no problems. (Green potatoes are technically poisonous, so green peels would be technically poisonous as well, but I've never seen it happen--either the chickens eat them or they don't, and what they don't eat will either dry out on the surface or be buried in the dark underneath layers and thus not turn green.)
Almost everything people eat is safe for chickens, so be sure to give the trimmings to the chickens. They like the peels, seeds, bug-bitten parts that the people don't want. Yes, I throw in things like onion peels too, which they probably do not eat--what they do not eat, they will compost.
Possible other crops for chickens and other birds:
slugs (I've read that ducks will eat bigger ones than chickens will.)