Is there any kind of immortality in plants?
I can find no examples of immortal plants, but then again immortality is rather hard to prove, it's rather like trying to prove that space will never turn in to a pony - as long as time exists it could still occur, only if you go beyond the end of all time would you be able to say that it has or has not happened but then time would no longer exist and nor would space/the pony. 

However, for examples of long lived plants it is perhaps best to look at trees - there are some species that live a long time and they can easily be aged. Here is a list of oldest known trees. It is difficult to tell how old these trees can become, and thus whether they are "immortal", because anything that lives that long is likely to encounter sources of extrinsic mortality during it's lifetime - predation, deforestation, extreme weather, erosion, natural disasters, contaminants, humans, disease, parasites etc. (rather than intrinsic dying of old age).

A lot of plants go through some kind of apoptosis, where cells are programmed to die. Plants such as annuals, biennials etc. die after a few months or years in a programmed way. It is difficult to say whether long lived species of trees would do the same after a few thousand years of life, because we just don't get to see many reach that stage of life, or if the mechanisms causing it have been suppressed (which could explain their long life).

Luke has posted a very thorough answer on the cellular and genetic aging of plants here. And to quote him:


  "...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"


Wikipedia has some very good pages on the evolution of senescence &amp; aging. This article has also a good section on plant aging and I hope you can access it.

and finally, if you want to have a laugh, take a look at this entry on plant immortality and aging from creationwiki.org...


  "Plants are actually or potentially immortal organisms that will live
  indefinitely provided their environmental requirements are met. In
  striking contrast to plants, humans and animals have a maximum life
  expectancy regardless of whether their needs are fulfilled. Although
  it is true that some plants grow as annuals, all plants are actually
  perennials by nature, and have adapted to a programmed life cycle in
  some instances to match growing seasons. Otherwise, plants only die
  because of biochemical starvation or disease. If their physical
  requirements are met, plants will grow indefinitely."


Some great clear science there, well done creationists... but it's just plain wrong.