here she is, folks!! i was gonna post before that we need antithesis, as the resident botanist, to give us the answer here. ...so are you gonna answer the question? and he meant shit as in poop.
And yes, plants die of old age. Many plants live for only one year and then die. Some plants have incredibly long lives and some kinds of orchids no one has ever seen die from natural causes...
some flowering plants have been deliberately bread to die of old age every year, these are called annuals. of course not all of them always do. perennials are of course supposed to not do so. the lives of individual plants may very even within their own species, just like any other kind of life form. unfertilized seads and spores and things like that, can remain viable indefinately until brought to life for the furst time. rhyzomes are a special case, that blure the lines between individual plant and sub-species. plants are not inhierently immortal, though some, such as the bristlecone pine, do live for a very very loooooooooooooooong time.
Some plants are essentially immortal. They do eventually die from environmental causes, but do not age or die from old age, i.e. there is no biologically programmed senescence. If you didn't age that doesn't mean you could never die, the probability of things like getting struck by lightning or a fatal car accident may be very low, but if you live long enough it's bound to happen, i.e the lines of the function of age and the function statistically predicting the next time you will be struck by lightning will eventually intersect at some point, possibly very far into the future.
this is an excellent article on the concept of immortality in some plants like the bristlecone pine that can live over 5,000 years! also talks about "immortal" animals like the hydra, and certain jellyfish. http://www.bbc.com/earth/story/20150622-can-anything-live-forever
A true virgin forest has lots of dead trees everywhere, standing up, leaning against other trees, and lying on the ground. As they rot away to nothing, they fertilize the ground, often resulting in perfectly straight rows of young trees thriving in the rich soil. And so, the life cycle wheel keeps turning. New redwood trees can sprout up from the roots of older trees, usually five or six feet away. Hundreds or thousands of years later, when the mother tree is long gone, her original location is made obvious from the circle of children that survive.