"I don't hope" is very rarely used, whereas "I hope not" is used
more often. If you don't hope something, use something like "I hope
[negative]." If someone says, "I hope [positive]," then say, "I
hope it doesn't." "I hope it rains tomorrow." [response] "I hope it
doesn't," or "I hope not. I want to go to the beach!" "I don't
hope" would imply that the person speaking never hopes about
anything, or never has any hope.