If telomeres become to short, they will trigger the cell's DNA damage detectors, which cause it to stop dividing and growing (i.e. become senescent).
If part of the DNA that codes for genes is damaged before this happens (or even after, due to complications with the process), the cell may undergo apoptosis—i.e., commit suicide.
As more and more cells in an organ enter senescence or undergo apoptosis, organs deteriorate.
Since telomerase does not act in somatic cells, this deterioration will get worse over time.
The process of organ deterioration with time is commonly known as aging.
Thus, the life span of organs—or even organisms—may be limited by their telomere length.
Experimental Evidence
A study done with the worm species Caenorhabditis elegans indicates that lengthening telomeres can extend life.
Another study found that increased insulin resistance (characteristic of diabetics) and weight gain were correlated with greater telomere shortening over time.
The telomeres of a certain long-lived bird species seem to lengthen as the birds grow older.
Telomerase and Cancer
Cancerous cells are found to have active telomerase in almost all mammals.
By over-expressing telomerase, tumor cells can replicate forever without stopping.
Without telomerase—like normal somatic cells—the tumors would eventually have to stop growing when their DNA damage detectors shut down replication after the telomeres were depleted.
The same over-expression is found in immortal strains of cultured cells.
In a way, there is thus a tradeoff: in return for a much-decreased risk of cancer, we are cursed with aging and its diseases.