'No touch' is a noun phrase - like 'some time' or 'any money' - a combination of a modifier and noun. It's sometimes used in product descriptions, for example, a lamp or a toilet flush which is activitated by simply waving your hand in the vicinity of the device. These products might be described as 'no touch' or as having 'no-touch technology'. 'Don't touch' is a command. It's a combination of a verb in its base form (touch) a negative auxiliary (don't) .This is an imperative form. We'd use this to tell someone not to touch something.

"No touch" is not really grammatically correct, but sometimes English-speaking parents say this to children to instruct them not to touch something. "Don't touch" is the correct way of asking someone not to touch something. However, sometimes you might see a sign (at an art gallery or a zoo, for example) that says "No Touching." This is also grammatically correct, but in speech it generally means "Don't touch anything," rather than "Don't touch this thing."