Have there been studies done to test Immortality of cancer cells in culture?
The HeLa cell line is undoubtedly the most used and investigated human  immortal tumor cell line. Extracted from a cervical tumor from Henrietta Lacks in 1951 at Johns Hopkins hospital, Baltimore, MD these cells proved immortal and are still used in many, many labs worldwide today. It is the oldest human cell line in use and, therefore, the oldest human tumor cell line (Callaway, 2013).

In terms of senescence of immortal cells; HeLa cells are immortal, and hence do not grow old as such. However, they are subject to mutations and the genetic profile of HeLa cells does change over time. Hence, although they may not be growing old, they do age in the sense that mutations are accumulating in continuous cell lines.

References
Callaway, Nature (2013); 500: 132-3