I've read that some people in some countries actually use tapeworms as a form of losing weight. What are the dangers to these people?
I haven't really found much on this topic (besides popular sites) but I can summarize it here: There are quite some tapeworms (or cestoda), I found numbers of up to 3500 species. They attach to the intestinal wall of the humans and then start to take up predigested food through their skin. With that, they reduce food from their host and start to grow, some get as long as 15 meters!
Some of the worms seem to be relatively harmless (besides stealing food), but this is more true for the first world. In poor countries, where there is not enough food, tapeworms can cause severe malnutrition.
Some tapeworms can migrate into the blood stream and from there into other tissues or organs like muscles, eye and brain. There they can cause cysts which can lead to organ failure and death.
An addition to the answer by Chris.
Probably the most dangerous tapeworms for humans are Echinococcus. In humans they form cysts (in a variety of organs including brain) [it's because humans are "interpreted" as intermediate hosts], which can become very large and can cause death when untreated or disrupted.