Yes, it does matter what faints it.
Destiny Bond only works on attacks that directly deal damage to the Pokemon using it (basically any attack like Tackle or Hyper Beam).
A Pokemon using Destiny Bond can not kill another Pokemon if it dies of Leech Seed, burn, poison, confusion, Curse (ghost version), Nightmare, Perish Song, and any other damage effects that are not directly caused by an opponent's Attack (meaning direct damage calculated by the BDF on that turn), but are rather once-a-turn effects of it (or, in Perish Song's case, when the perish count is 0.)
As for Perish Song/Destiny Bond, Missy excels in using this for a suicide mission. It Mean Looks, Perish Songs, then proceeds to either Protect or Destiny Bond. Destiny Bond is useful for taking out major threats like Curselax that Missy (or the whole team, for that matter) will not survive.
- - - - -
"Young love, with horns!"
-Toonami review of Ico (PS2)
[Rockzilla] Remember now, excuses are like asses, everybody has one and they all stink.