mp3 players use flash memory to store the files. My recollection is basically what has already been mentioned.. i.e. it is the writing to the flash memory that is the limit to its life time. Reading from the flash memory isn't hard on it.
On a slightly different tack... the bigger question is about what will eventually cause the player to die.
My personal experience has been that my little mp3 players become obsolete about the same time that the switches start to fail. The latest is a great little Sansa Sandisk Clip mp3 player and FM tuner. It's a wonderful tiny gadget, but the switch contacts are wearing out and the lithium-ion battery capacity is falling quickly.
Steve in Peoria