Cancer biology: Can tumors form from cells that are genetically identical to non-cancerous cells?
This is a very interesting question, and one that has been at the heart of cancer research for a long time. I think it separates into three issues.

Are there non-genetic mechanisms that can induce uncontrolled cell proliferation?

Yes, at least in experimental settings. There are many viruses that induce proliferation in its host cell, and viral genes / proteins have been identified that can disable the host cell's cell cycle regulation machinery. A well-studied example is the SV40 Large-T gene (from Simian Virus 40); this gene can be expressed in cells to cause cell proliferation in a transient, reversible manner, and it does so by interacting with host proteins, without causing genetic changes. See for example this article.

Can such genetically normal cells progress to form tumors?

It depends on what you mean by "tumor". To my knowledge, tumors large enough to be a serious medical problem are always found to be genetically altered in one way or another. There are plenty of viruses that cause tumors in humans, for example Epstein-Barr and papilloma (HPV) virus, but in all cases viral sequences are found integrated into the host genome. This may be because development of large tumors requires many cell division, and any non-genetic change that drives proliferation is likely to be lost on the way.

On the other hand, HPV virus frequently causes warts, which technically are small benign tumors. As far as I know, no genetic modification of the host cell is required for this to happen; rather, the virus replicates from its own separate DNA while simultaneously driving keratinocyte cell proliferation to cause these growths. So this example shows that small "tumors" can be formed without any genetic cause. Perhaps there is a limit on how large tumors can grow without genetic changes.

Can genetically normal cells form cancer, meaning a malignant (invasive) tumors?

No, as far as I know, this has never been observed. Large, malignant tumors without genetic changes are unlikely for the reasons above. There are cases of metastatic cancer where the primary tumor is so small that it cannot be located, but in these cases the metastatic cells are again genetically altered. Experimentally, it is possible to produce invasive behavior in cell lines by expressing oncogenes like ras, but these genes are mutated to begin with. Also, normal cells have mechanisms that detect such alterations, causing senescence, and they must first be inactivated by mutations. Transformation to an invasive phenotype is not usually caused by viruses, probably because it would give no advantage to the virus (unlike cell proliferation).

For more information, Robert Weinberg's Cancer textbook does a good job of discussing these issues. This article also provides a historical perspective of tumor virology with some interesting pointers.