Some types of allergies affect people when they touch certain substances. They then break out in a rash. Such an allergy is called allergic contact dermatitis, and it's the result of an immune system mistake. When your body detects that you've touched something that it thinks is dangerous, it sends out antibodies to deal with the situation. The antibodies then trigger a bunch of chemicals, one of which is histamine. The histamine is responsible for your allergic symptoms, rashes included.
Some people get allergic contact dermatitis when they come in contact with the leaves of tomato plants. The result is extremely itchy red patches, hives or bumps, and these symptoms can take days to develop after you touch the plant. However, there can be a variety of causes of allergic contact dermatitis, and your best bet is to let a doctor determine whether tomato plants are the true cause of your rash.