Eddie died alone on the tracks, but wasn't granted the release that physical death would have provided. His sin, as stated, was that he lived for another man (woman/Dagny) not for himself.
Rand ruthlessly demonstrates what happens to followers when there is no one left to follow. He ended up as the dust within the hollow tree. I quote for reference...
"The great oak tree had stood on the hill over the Hudson, in a lonely spot on the Taggert estate. Eddie Willers, aged 7, liked to come and look at that tree. It had stood there hundreds of years, and he thought it would always stand there...
...He felt safe in the oak trees presence; it was a thing that nothing could change or threaten; it was his greatest symbol of strength.
One night, lightning struck the oak tree. Eddie saw it the next morning. It lay broken in half, and he looked into its trunk as into the mouth of a black tunnel. The trunk was only an empty shell; its heart had rotted away long ago; there was nothing inside - just a thin gray dust that was being dispersed by the whim of the faintest wind. The living power had gone, and the shape it left had not been able to stand without it."
Good catch. I hadn't picked up on the tree being an analogy for Eddie's life in particular. But, that one line does make it clear. In the end, that's exactly what happens to Eddie.