False hope is hope built entirely around a fantasy, a hope that has no knowable chance of coming to fruition. The exploitation and creation of false hope in other people for material gain is at the core of many of the scams covered on this site. It can be found in quack medicine, politics, religion, economics, law, and many other fields.
Creating false hope in others is a particularly heinous act. Often the last refuge of the crank or promoter of pseudoscience will be "at the very least, what I do doesn't harm people" -- but harm is exactly what a peddler of false hope creates.
 False hope in health
False hope is abundant in issues revolving health. Health serves as a fertile ground because it is so salient to our daily lives. Everyone wants good health, but often can not or do not want to do the things necessary to have it (and sometimes there is nothing that can be done). Something that everyone wants, but not everyone can get is easily manipulated by quacks and frauds selling their easy non-solution solutions.
At its most extreme, peddling false hope in health revolves around pretending to have cures for deadly and incurable, or difficult to cure, diseases. This is where the slimiest and vilest people thrive, offering up quick and easy cures for cancer, HIV/AIDS and autism. Others claim to have found secret causes for these diseases, such as vaccines, or completely deny the germ theory of disease, or push failed pseudoscience like homotoxicology.
Less extreme are the peddlers of easy weight loss schemes with no dieting or exercise, worthless cures for the common cold, and many other more transitory or life style health issues. The major players in this area are homeopathic snake oil, bodybuilding woo, and questionable herbal supplements.
 False hope in religion
Religion at its core is the personification of false hope, such as the promise of immortality and bliss, or the 70-odd virgins, in heaven. But one doesn't even have to get into these deeper examples of false hope to find many more crude examples.
A favorite scheme of televangelists is to sell financial security. Basically they tell you to send them all your money and God will find a way to give it all back -- times ten! Some of the more stubborn pushers of this false hope are Pat Robertson with his law of reciprocity and Peter Popoff with his magic salt.
Another nasty false hope peddled by these charlatans is faith healing. Grotesquely personified by the likes of Benny Hinn, con-artists have made veritable fortunes pretending to magically cure everything from paralysis to cystic fibrosis by "laying on hands" or merely mumbling to themselves. Several whole religions revolve around this non-existent power of faith to heal, such as Christian Science, whose practitioners regularly cause their offspring to die horrible deaths from normally curable disease.