That is a really great question. Yes Pokemon can die. They can die of illnesses, hunger, and old age. But unlike animals, they can die when their trainers die. A Pokemon can die especially when it is abandoned and left alone for a long time. There are some entries in the PokeDex that can talk about Pokemon dying. Here, I'll lay them out for you:
The flame on Charmander's tail burns bright from birth. It is said that if a Charmander's tail flame ever goes out, it will die.
Spoink has to bounce to live. When it bounces, the bouncing keeps its heart pumping.
So, yes Pokemon can die. There are a few Pokemon that died during the series. Such exceptions are: Latios from Pokemon Heroes, Lucario from Lucario & The Mystery of Mew, and the Stoutland from the Sun & Moon series.