I would say earbuds have their place in everyones collection, be it a stock earbud that came with your music player/phone or something you might have invested in for better quality. For some people, they don't like the feeling of something sticking in their ear (or may never have tried/don't like the idea), the earbud isn't a bad idea in that situation and for portable use.
The main downside I believe being a lack of isolation in noisy environments, where one would need to turn up the volume significantly to hear the music, this for say 1-2 hours a day can damage your hearing.
Occasionally, I would use earbuds at night laying in bed, there's no increased pressure in my ear canals and I can truly relax to some music. Plus you also have additional degrees of freedom (for head movement while laying down) with something smaller than a headphone.
But to answer the question proposed, I would say, yes and no. Aftermarket earbuds are slowly dying but they are not dead yet. We can observe the changes in the market to realise this, comparing the proportion of in-ears to earbuds. But as long as MP3 players and ipods/iphones continue to bundle earbuds, I would say they have some healthy years left.