By clogging the main blood vessels, the blood supply to other organs of the body is reduced, particularly the lungs, liver and kidneys.
Of course, much more work for the heart results. Therefore, tachycardia (rapid pulse) and dysnea (shortness of breath) are typical.
Blockage of major blood vessels can cause the animal to collapse suddenly and die within a few days.
Sudden destruction of large numbers of mfs occasionally causes severe shock-like symptoms that may kill the dog.
Most dogs can be treated successfully if heartworm disease is detected early by blood tests. Other diagnostic tests include radiography and electro/echo-cardiography.

Human filariasis, mostly in Africa and Asia, is discussed in Filariasis: http://www.pitt.edu/~super1/lecture/lec10971/index.htm