I've been doing some reading, and have come up with the following interesting information.
During cell division the DNA is replicated, but the mechanism is imperfect and in each round of cell division a small section is lost from the end of each chromosome. To compensate and protect the genetic information there are caps – regions of excess nucleotides – at the ends called telomeres (Silvestre, 2012).
Mammalian cells do not replenish their telomeres after cell division, giving them a replicative lifespan; a maximum number of cell divisions before the telomeres run out. The cells then either enter a state of senescence and no longer divide, or initiate programmed cell death. However each cell has the genetic information for telomerase – the enzyme that can replenish the telomeres – but cannot use it. Only embryos and stem cells are able to express telomerase in mammalian cells - this protects against the build up of damage to DNA causing uncontrolled cell cycle progression. Activated telomerase is a core feature of many cancers (Campbell, 2012).
You mention telomeres in your question, and it is an interesting question I had never considered. Plants have no germline preserved, so their progeny must bear all the defects of their particular parent cell, and that plant cells must therefore all have the capacity to regenerate their telomeres. I found a paper by Fajkus and colleagues that finds telomerase activity in plant cuttings grown in culture (Fajkus, 1998). Their hypothetical Plant Telomere-Length Regulating (PTLR) protein must arise in the early stages of the culture and last only a few cycles. Interestingly it seems that the efficiency of this process varies between different plant species. The authors note that their plant of choice – tobacco – is particularly proliferative and that other plants such as barley were harder to grow from cuttings, which may be down to a different way of regulating the telomerase (for instance it could only be activated occasionally at low amounts).
Do Plants Get Cancer?
It seems to me the next obvious question. One of the first papers I came across (published in 1916 by the Journal of Cancer Research!) talks of Crown Gall in plants and its relation to human cancer (Smith, 1916). Many websites cite galls as being the equivalent to cancer in plants, but it appears that the outgrowths (galls) that appear very tumour-like are in fact caused by bacteria encouraging their surrounding cells to proliferate. (See this question for more detail on galls).
I later found an abstract (I cannot access the full paper) from a paper published in Nature in 2010 called “Walls around tumours — why plants do not develop cancer” (Doonan, 2010). It appears that Plants can and do develop tumours, but they are less frequent and less lethal due to fundamental differences in development between plant and animal cells. Singh and colleagues found that in 2 plant species (Arabidopsis and rice) there were DNA damage repair pathways well conserved but with variation; there were several gene duplications in different DNA repair pathways (Singh, 2010).
Plants Do Not “Age” As We Do
Although I could find no conclusive evidence, it seems to me likely that plants may also have enhanced maintenance against other cellular stresses than DNA damage, such as build-ups of aberrant proteins. It therefore appears that in a protected environment some plants could live indefinitely. I say some, because others deliberately end their lives after reproduction. Traits such as longevity are inherently hard to be selected for, as the advantages come long after reproductive maturity and are therefore under much less selective pressure.
We see plants wither and die, but this is because they are constantly subject to environmental insult. Animals and insects may eat them, there may not be enough food or water, or sunlight, or the plant may become infected. All of which are at the macroscopic level. Due to completely different evolutionary constraints and pressures plants are highly resilient to cellar stresses, possibly due to differences in metabolic rate, and apparently have indefinite replicative potential.
Although the more I think about it, the more it seems that most plant must have an ultimate lifespan, in that as they forever grow and expand their tissues would gradually harden and become less functional (I am mostly imagining bark plants here). Although as the layers of bark are pushed out the layers below take the place. In this way a tree could possibly live indefinitely by continuously remodelling it's layers.