The simple answer is yes, all plants die. Plants are very different than we animals are, though, and the answer to your question is actually not as simple as that.
First, plants have what's known as "indeterminate growth." That means that,unlike animals, there's no set size or age when a plant is considered mature or old. If conditions are right, they can simply grow and grow with almost no limitations. (The two limitations they'd eventually experience would be that they'd get so big that 1. they could no longer support the weight of their own bodies, and 2. water could no longer travel reach all the way from their roots to their branches.) If humans and other animals were like this,it would be like you being 25 feet tall and still growing as long as your parents kept feeding you!
The second big difference between plants and animals is that most plant cells can change at any time into another cell type, dividing many, many times in the process. This is called being "perpetually embryonic," and it's why plants can keep growing indefinitely. It's also why you can stick a leaf or green twig in a glass of water and it will start growing roots. Animals have very rare and special cells like that called stem cells, but most animal cells are stuck being what they are (like skin cells or nerve cells),and they can't really divide much anymore.
What all this means is that the reasons why animals die (we stop growing,our cells stop dividing and "wear out") don't affect plants. Many kinds of trees commonly live for thousands of years. Other plants send up new plants from the end of their roots--those new plants do the same thing, and this continues on and on until you have a single plant that's many miles across!It looks like many different plants to us, but that's just because we can't see that they're all connected underground. The first part of the plant in the middle may eventually die, but all its other parts keep living and living--I don't know if anyone knows how old those plants can get.
Just because of the laws of physics, though, I think it's safe to say that even those plants eventually die. Plus, you have to remember that conditions in nature are never perfect for very long. So, even though a plant has the ability to possibly live forever, something will eventually happen to kill it, like drought, fire, disease, or other plants using up all the soil nutrients.
Excellent question!