If a cancerous tumor has a lot of mutations in them why can't the immune system detect them? If a person has cancer could this somehow alter the person's immune system so it doesn't function effectively to detect greatly mutated cells? Does the fact that a tumor is 'made' of 'mutated' non-foreign human cells help it to be undetectable?
This is a good question and it gets to the most basic foundations of immunology. Your immune system is made to be flexible - able to adapt to almost any challenge. However, with so much flexibility comes the potential to generate immune cells that react against the 'self'. So, at the other extreme, the immune system needs to be limited.
In order to meet these two (potentially conflicting) goals, immune (at least lymphocytes) cells have a two-step development pathway. In the first step, the body randomizes its immune receptor specificity. In the second step, it ensures that these receptors are not auto-reactive.
Doing so means that immune cells are aware of their environment at the time that they are 'born.' Take a look at this review from the Fuentes-Panana lab to read about how tumors actively cultivate the immune system of their host in order to maintain a more favorable environment.
Great question and in part, you are correct. However, you must also remember that this greatly depends on what genes have undergone mutation.
In the case of cancer, it's usually either a gene that regulates cell proliferation rates, induces programmed cell death or helps to repair damaged DNA. When one (or all) of these genes is mutated, the cell constantly proliferates and/or does not die when it is "programmed" to do so or cannot repair mutations to it's DNA, thus creating a cancerous tumor.
Now, on to the immune system. The reason a persons immune system does not target and kill it's own cells is because of a certain set of genes that is specific to each and every person called Major Histocompatibility Complex (MHC). These genes encode specific cell surface molecules that the immune system cannot produce antibodies to bind to because they are "locked", if you will (see, the immune has a sort of built-in list of genes that code for antibodies that bind to different sequences of molecules (that's how you develop immunity and fight pathogens), the genes that code for the specific molecules that your cells have on their surface are just not on that list.) In order for a person's immune system to recognize cancerous cells as a pathogen and destroy them, the cancer cell's MHC genes would also need to be mutated drastically.
For instance, breast cancer is usually caused by a mutation in the BRCA1 or BRCA2 genes. These genes assist in DNA repair. Even if this mutation happens, these cells still have the same MHC as the rest of the cells in your body because BRCA1 and BRCA2 don't really have much to do with MHC. Because of this, the immune system does not recognize the cancerous cells as a threat and is unable to destroy or even detect them.
I hope this answers your question.