Can lymphedema be fatal?

Rarely: It is impossible to give a specific answer. It would depend on what other conditions coexist with the lymphedema. Severe lymphedema could coexist with cardiovascular disease, diabetes etc. In severe caeses, ulceration , infection can occur and lymphedema can become permanent.

Not directly: Often times very careful skin care is necessary for sufferers of lymphedema. If skin breaks open and becomes infected badly enough, a blood borne infection could develop. It is a bit of a stretch but, in that case the situation could conceivably become fatal.

Minimal association: Lymphedema is when there is compromise of the lymphatic circulation. This third circulation (arteries, veins) carries things too thick to be safely carried in arteries and veins like bacteria eaten by white cells and digested food. It's pushed along by normal body movements. It increases risk of infections and reduces mobility which contribute to overall risk of death but by itself, not really.