Firstly, a tumor's own blood vessels might rupture or it might destroy other blood vessels during its growth, leading to severe hemorrhaging, which in turn can cause the patient's death.
Secondly, tumor cells may detach and enter the blood circulation, forming emboli. If these emboli cause blockages in critical organs, such as pulmonary embolism, they can lead to death.
Thirdly, the growth of the tumor consumes a lot of nutrients, causing the patient to develop a cachexic condition which lowers the immune system and makes infections and other diseases more likely.
Thus, the mechanism by which tumors impair health is through the over-replacement of normal cells with inferior ones.
The process by which tumors impair health and lead to a patient's death can be likened to the collapse of a company.