What is the disadvantage of having no anticoagulation after pulmonary embolism?
Not always, however typically it is given for at least a few months. Lack of anticoagulation therapy would greatly heighten the risk of having another PE or related cardiovascular events. At the worst side this could cause death or permanent disability, and PE related deaths are quite common. However this depends on the nature of the PE and risk factors. For example, obesity, cancer and the pill are all risk factors and if these were reduced then it is likely the person wouldn't have another PE despite not being on long term anticoagulation therapy. However, if there is an underlying cause even despite anticoagulation therapy a person may suffer a PE. 

The problem with anticoagulation therapy is of course it stops your blood clotting. Thus haemorrhages are more common and also more severe. However in an individual receiving anticoagulation therapy, they typically have increased clotting so this may be balanced out by therapy.

Finally anticoagulation therapy is typically warfarin in the long term. Warfarin interacts with so many drugs including even antibiotics, it is teratogenic and quite sensitive to diet (any changes in Vitamin K will directly affect the effectivity of warfarin) thus frequent monitoring is required. This is annoying for the patient as they have to report for frequent blood tests as well as watch their diet etc.