"I hope not" means there is a situation that you wish not to occur/ to exist. Something like this "Are the zombie penguins still attacking our village?" "I hope not.. I'm scared"
"I do not hope" refers to the act of hoping, and that the person does not hope. "Hopefully, the penugins are gone!" "I do not hope - I take action. Let's kill those zombie penguins"
Both meanings could be expressed by the German "Ich hoffe nicht", if you pronounce "hoffe" more strongly it means the 2nd option, if you pronounce "nicht" more strongly or do not put particularly stress any of the words it means the 1st option.