The most important word in your question is "conclusively". Because most studies involving the subject are performed using various model systems (both in vitro and animal models), it is likely to be difficult to "conclusively" prove a strong relationship among various topics (telemeres, telomerase, cancer, and aging). However, there is ample evidence to support the idea that (i) immortalization is the first step for cancer cells; and (ii) defects in telomere that lead to activation of a DNA-damage response (DDR) contribute to premature aging in cells/organisms.

Divaker Choubey, thank you for your informative response, which I take as a confirmation of the puzzle before us.  On the one hand, if I understand correctly, natural or induced telomerase would present a hazard of immortalizing cancer cells and perhaps even of premature aging.  On the other hand, we have a report of "the first human being to be successfully rejuvenated by gene therapy, after her own company’s experimental therapies reversed 20 years of normal telomere shortening" (cited above). 

While the former possibilities and the latter reported reality are not logically mutually exclusive, the appearance that they coexist seems to create more surprise than is consistent with literature known to me.  For example, in the last-cited article from researchers at the University of Utah comes the statement that scientists "have been able to use telomerase in the lab to keep human cells dividing far beyond their normal limit, and the cells do not become cancerous."

We have also forward-looking conclusions in the research-literature based on human cell clone testing; e.g., "Thus the ability to extend cellular life-span, while maintaining the diploid status, growth characteristics, and gene expression pattern typical of young normal cells, has important implications for biological research, the pharmaceutical industry, and medicine" [Andrea G. Bodnar, et al., Extension of Life-Span by Introduction of Telomerase into Normal Human Cells] Science, Vol. 279, 16 Jan 1998].

But a human being is much more than a cell.  What seems to be absent in the literature is a testable hypothesis concerning the aging effect of introduced telomerase in animals for which the hazards you mentioned are known and applicable.

Dear Albert,

to be honest, I'm not sure if there is a chance for the relationship that you postulate. These phenomena (cancer and ageing) are too complex to be related exclusively to telomerase and telomeres, respectively. Moreover, please note that several types of cells senesce with maintained telomere length.  

Thank you, Krzysztof, and perhaps I might clarify that cancer is in this mix of potential correlatives only because it seems to have been argued that telomerase introduction appears to pose or increase a risk of immortalizing cells that pose a danger to healthy aging. Perhaps a future respondent here will enlighten us by citing an accessible paper describing research that, in the words of Andrews and Cornell, would show that "a pharmaceutical telomerase activator would be more likely to prevent cancer than to cause it."