If somehow a human could give a tumor unlimited resources, would the cancer grow forever? It seems like it would until it gets so large that it physically affects vital organs. Is what would likely happen?
Yes. The very first cells used to study cancer are still around (HeLa Immortal Cells - Named for the subject Henrietta Lacks) and are basically immortal as long as they're fed.
As for tumors, whether cancerous or not they most definitely can continue to grow until they become a serious medical issue (WARNING: GRAPHIC - 3 Largest Tumors Recorded). One of the largest recorded tumors was 300lbs (~140kg) in a woman's abdomen that sprouted off of an ovary. They had to operate with her laying on her side, otherwise it would have crushed her abdomen and killed her.
Not all forms of cancer are immortal or produce large tumors (the 300lb tumor was benign), but it's definitely within the realm of plausibility.