Anaemia from blood loss
Some tumours, particularly of the digestive system, can bleed so slowly that the bleeding itself may not be obvious. If it continues over a long period of time then eventually the person becomes anaemic and goes to the doctor feeling tired.
This happens if the rate at which blood is lost is faster than the body's ability to replace the red blood cells. Healthy bone marrow produces the equivalent of about 20ml of fresh blood daily - over 70 litres per year - to replace the normal needs of the body. The limiting factor in long-standing blood loss is usually the availability of the raw materials for the bone marrow to work with rather than the capacity of the bone marrow to respond. Iron is an important component of the oxygen-carrying molecule called haemoglobin, which is contained within the red cells and a lack of iron therefore holds back the production of new, properly formed red cells.

Anaemia due to iron deficiency is the characteristic finding with a cancer of the colon (large bowel), stomach or gullet. Often the anaemia is the only clue to the presence of a bleeding source somewhere. Any person who develops iron deficiency anaemia with no obvious cause should be investigated for the presence of a bleeding point within the digestive system. Fortunately not all bleeding sources turn out to be cancers.

Of course any source of regular or prolonged bleeding can possibly cause iron deficiency, so a check should also be made for blood loss in the urine or, in a woman, from the womb although both of these sources are more likely to cause obvious blood loss.

As we depend on the iron in our diet to supply us with our needs, iron deficiency anaemia will occur more quickly if a person's intake is poor for any reason or the quality of their diet is unsatisfactory.

For example, someone with a tumour of the gullet or stomach might have difficulty swallowing a normal diet, so a supplement of vitamins or minerals is sometimes necessary to add to what they can take. Cancer can be associated with reduced appetite, which can add to a person's nutritional difficulties.

Anaemia due to bone marrow failure
The bone marrow has a rich blood supply, and is the site of manufacture of all blood cells. If a cancer in one part of the body sheds some cancer cells in to the circulation, which is often the way cancers spread (metastasis), there is a good chance that the circulating cancer cells will enter the bone marrow and lodge there. The bone marrow is therefore a common site for a metastasis to develop. Cancers of the breast, prostate and lung are the commonest type to do this although almost all cancers have this capability.

Once in the marrow the cancer cells can multiply easily, being well supplied with blood. The tumour deposit enlarges, occupying more and more of the marrow space, so reducing the amount of blood-producing marrow. It does require a large amount of marrow to be replaced with tumour for this alone to be the cause of anaemia, as the bone marrow has a fair amount of reserve capacity. However, someone who has widespread metastasis will usually be in the advanced stage of a cancer, in which the body's metabolism seems to be impaired for reasons that we do not yet fully understand. In such circumstances anaemia is more likely to occur.

There are some tumours that arise from the bone marrow tissue itself, such as some types of leukaemia and multiple myeloma. (link) As these are more directly involved with the bone marrow's function they are more commonly associated with anaemia.