I don't think false hope is inherently bad. It's when you build your whole life around this false hope that it becomes futile and dangerous. Jehovah's Witnesses do just that. They hope for the better future. They hope God will create a paradise earth for them. The reason why it's a problem is that they build their lives around that. Sometimes, they reject a job promotion because they think it would interfere with their spiritual goals. Sometimes, they deny themselves the opportunity of getting a higher education for the same reason. Sometimes, they don't accept blood transfusions, again for the same reason. That is quite obviously a problem.
But let's say that there is a "worldly" person who learns that she is terminally ill and most likely will be dead within a year. Let's say that she also enjoys reading books—like really enjoys. In addition, let's imagine that there will be a new book coming out in one and a half years, and she really wants to read this book. Her hope that she will live long enough to read it is in some sense "false." According to multiple doctors, she won't survive for more than a year. Is it necessarily bad if she hopes to see the book one day—if this is one of the things that keeps her going?
There is of course another thought experiment that applies just as well. Is having no hope worse if that very same person, who happens to be an atheist, lives in a region where she has the legal right to die at any time (because of her terminal illness)? Is it worse if she also believes she won't go to hell or won't be punished in any other way for choosing to die? Is it worse if she also believes that she doesn't have to fear death because she doesn't believe in the afterlife anyway? Some would say that in such a case, choosing to die because having no hope is better. Why suffer a year of some terminal illness if you can die already without experiencing the suffering?
So is false hope bad? Sometimes it is, sometimes it may not be. Sometimes it is for some people, sometimes it may not be for some people.