In the Mistborn series does anything unique happen when you die? We have a few glimpses of people after they die in the series so what if anything happens?
Most of this answer will be drawn from information revealed in Mistborn: Secret Histories, so consider this a spoiler warning for that novella.
Whatever is left passes back through the Realms. Your spirit, or soul, or whatever else is left and vaguely defined, is dumped into the Cognitive Realm, which can be considered a shadow realm to the Physical. From there, it quickly (as in, seconds, rarely up to a minute) dissipates as it is inexorably pulled onward, into an unknown beyond, supposedly coming to rest in the Spiritual Realm. Further glimpses of Realm theory suggests that all souls are projections from the Spiritual Realm tied to a Physical body, and once that tie is cut the soul just slides back to its source, the Cognitive serving as the "in-between" for the journey.
Beings of greater Investiture can hold themselves together longer. Souls still have will. A choice can be made to let go and pass on, ending the journey in the Cognitive immediately, or to fight for their continued existence, which does not mean much against the natural forces involved but can buy some extra seconds. Full mistborn are specifically called out as being able to hold onto their "grip" in the Cognitive, although this is really just a matter of some extra minutes. One soul with immense power travelled a few minutes from his point of death without fading even slightly, and made the decision to move on shortly after. Presumably other souls of power (like Feruchemists) show the same effect. Investiture also seems to be a projection of the Spiritual, although one of energy rather than mind and heart, so this ability may just be a "push" on the natural transit, slowing down the journey of the soul.
There are living beings either existing in the Cognitive, or travelling through it , that can potentially interfere. Shardholders can easily appear to the dead as their souls manifest, and offer guidance, comfort, or in at least two cases, a choice about what happens next. In theory anyone travelling the Cognitive could do the same, although their power to offer choices would be suspect, and deliberately finding a particular soul in transit would be beyond difficult due to timing (matter of minutes) and the very ephemeral and flexible nature of the Cognitive.
To bottom line it, in Mistborn, characters appearing after they are supposed to be dead are a sign of the last point, interference. At least one soul was returned to life after dying (although this was assisted by power in the Physical), and at least one soul was preserved as a ghost (in technobabble, a Cognitive shadow) after trying desperately not to move on, just in the Mistborn series.
Other Cosmere novels like Warbreaker (the Returned) and Elantris (the Elantrians have a lot in common with Returned) may contain further examples of interference. Then again, notes from Khriss in Arcanum Unbounded imply that this natural order of the Realms isn't a constant, and something catastrophic enough can have severe consequences and unnatural effects. The Shades in Shadows for Silence in the Forests of Hell may fall into this category, or even the Elantrians, depending on interpretation.