No. Just no.
If you die in a dream, you do not die
in real life. It doesn't happen. I've done it myself, everything from
plummeting to my death to drowning, even having the dream starting with
me dead. I'm clearly still here.
Dying in a dream isn't even always a trigger to wake up. More often than not, it seems, the dream continues after you've "died" in it, whether you're a ghost, a reincarnated being, or find yourself in some sort of afterlife. Also, however you continue the dream may not necessarily reflect your personal views on the matter or indicate any degree of doubt in your personal views.
Worth noting as well: experiencing death in a dream does not automatically classify it as a nightmare. Rather, death is a mere consequence of actions that may be inherently positive.
Besides, if someone
died in a dream, and then died in real life, how would you know? Though
the techniques and technology do exist to determine what someone is
dreaming, they are not used on a broad enough scale to determine that
people actually do die in their dreams.
I think the
more appropriate question is: if you die in real life, while dreaming,
what happens then? Does your soul continue dreaming? Do you then die in
the dream? I know that the dream does not physically affect the waking life, my question is if the reverse is true.