This is the question posed and not answered by a recent paper from a German institution, accompanied by a very tersely worded abstract:
The process of aging remains a great riddle. Production of reactive oxygen species (ROS) by mitochondria is an inevitable by-product of respiration, which has led to a hypothesis proposing the oxidative impairment of mitochondrial components (e.g., mtDNA, proteins, lipids) that initiates a vicious cycle of dysfunctional respiratory complexes producing more ROS, which again impairs function. This does not exclude other processes acting in parallel or targets for ROS action in other organelles than mitochondria.
Given that aging is defined as the process leading to death, the role of mitochondria-based impairments in those organ systems responsible for human death (e.g., the cardiovascular system, cerebral dysfunction, and cancer) is described within the context of "garbage" accumulation and increasing insulin resistance, type 2 diabetes, and glycation of proteins.
The processes involved in mitochondria-based impairments are very similar across a large range of organisms. Therefore, studies on model organisms from yeast, fungi, nematodes, flies to vertebrates, and from cells to organisms also add considerably to the understanding of human aging.
Would we age without mitochondria? Certainly. Mitochondrial damage and dysfunction is only one of the most likely fundamental differences between young tissue and old tissue. Other similarly fundamental differences include the build up of amyloids, lipofuscin, and cross-linked proteins (the glycation of proteins mentioned in the abstract above), and there is no reasonable case to be made for those aspects of aging to be dependent on mitochondrial function. If all our mitochondria were removed and replaced with black box nanomachines that performed the same tasks for an indefinite period of time without wear or failure, then we would still suffer degenerative and eventually fatal aging due to the other forms of accumulated damage that harm our tissues.
(Replacing our mitochondria with something that doesn't fall apart and progressively sabotage us is actually a very plausible and desirable goal for the future of medicine. But the first strides in the that direction will probably take the form of simple replacement: introduce new mitochondria or new mitochondrial DNA to take up the slack).
Why do we age at all? There are plenty of species that age far less visibly than we do, suffering little reduction in vigor and resistance until very late in the game. But they have mitochondria too, so I don't think we can lay the roots of aging at the feet of mitochondrial biology. If there were no mitochondria, then the forces of natural selection would still result in the vast majority of species consisting of individuals that age to death. The only differences would be in the details of how eventually fatal damage arises and progresses. The world changes, and that favors aging as an outcome of successful strategies in evolutionary competition.
When conditions change, a senescent species can drive immortal competitors to extinction. This counter-intuitive result arises from the pruning caused by the death of elder individuals. [While] senescence damages the individuals and has an evolutionary cost, it has a benefit of its own. It allows each lineage to adapt faster to changing conditions.
This is all natural, but not good. It is natural like anthrax, dying of exposure, plague, and famine. Our ancestors toiled hard to defeat the aforementioned line items, very successfully in recent centuries, and research and development efforts continue today. Here and now we can also choose to do something about the causes of aging, an option that wasn't available until the present time: mitochondrial damage, amyloids, and the other causes of aging can all be addressed with the tools of the biotechnology revolution. So we should get on with it, and those of us without the necessary skills to work in the laboratory should help to fund those who do.