I can't really tell you, since when a Tamagotchi dies, I usually am not near it.
All I know is that, on the Connections, they die if you doesn't feed them for a long time or you doesn't use medicine when they're sick. I think that's the major cause of death on the Connection, since the character, after a while you doesn't feed them when they are hungry, call your attention with special animations such as angry moves.
I heard that a baby cannot die on the Connection, even if you treat him/her really bad. He just wait to become a child.
That's all I know.