The Lamp-post was a major landmark in the country of Narnia, located in the north-western area, populated by Dryads and Fauns, which was named the Lantern Waste after it. Resembling a London streetlamp, it stood in the middle of the forest and shone day and night. It was at the lamp-post that Lucy Pevensie first met Mr. Tumnus, who told her that it marked the beginning of Narnia.
The Lamp-post originated on the first day of Narnia's creation, from a bar of iron that Queen Jadis (the White Witch) had torn from a London lamp-post, which she had thrown at Aslan. Aslan, at the time, was creating the living things of Narnia through his song, which made the ground of Narnia magically fertile, and gave birth to all the animals and plants of the world. The iron bar fell to the ground, and, under the influence of Aslan's song, grew into a new and full-sized lamppost in just a few hours.
The Lamp-post's ability to burn continuously without fuel (old London streetlamps ran on gas) may have been due to the fact that it is an organic living thing, not a manufactured artifact, as other street-lamps are.
The ultimate fate of the Lamp-post is unknown. It is possible that it endured through the ages of Narnia's existance, and continued to shine until all the light went out, and Narnia came to an end. This, however, seems rather unlikely.
Most of the action in the last days of Narnia took place in the area of Lantern Waste, which is where the lamp-post was located. The lamp-post was never mentioned, and, apparently, never seen. This makes it highly unlikely that the lamp-post still existed by that point in Narnia's history.
It may have been extinguished or destroyed by invaders, such as the Telmarines or the Calormenes, after the Pevensies left Narnia. It is also possible that, as it was a living thing that grew from the ground, it may have eventually died.