This is a very hard question to discuss because it depends, as so much in philosophy, on definitions and context. What is false hope, what is no hope, and how do we prove them? Is the context eschatological (or existential, if one prefers) or routine?
The issue of context is most easily dealt with. Hope obviously has a different and more important effect in the former context (“I hope there is an afterlife”) than in the latter (“I hope the grocery store has my favorite food”). Consequently, it is important to confine arguments to their proper levels.
Ancient texts have multiple views. Some (I don’t remember the sources now) regarded no hope as the worst punishment, and this was taken up and most famously expressed by Dante in Canto 3, “Abandon all hope ye who enter here.” Hope certainly nurtures the human spirit, as can be seen in most real-life survival narratives, so I would expect the loss of all hope to be absolutely crushing to the spirit. On the other hand, the myth of Sisyphus implies that false hope is worse. The question is further complicated by the strong human tendency embodied in the adage, “Hope springs eternal.” Camus notwithstanding, I am unfamiliar with the source material for the myth of Sisyphus. However, I wonder whether Sisyphus continued due to hope or because he was Zeus’ marionette. If the former, was he really worse off, or simply an eternal, admittedly frustrated, optimist?
Cicero also weighed in on this question (in his essay, “On Old Age”) by arguing that death was not to be feared because if there is no afterlife then death doesn’t matter, and if there is an afterlife, then death is not fearsome if one has lived a good life. This highlights that “false hope” really can’t be determined in an eschatological context.