When the perl interpreter stops because die was invoked, your reference goes out of scope. So no matter what, the object is being destroyed because it went out of scope - see Destructors. The question you want to ask is "How can I detect in a DESTROY method whether die was called?"

A better question I should be asking is "Why do you need to know?" Is this just for near-term debugging purposes or do you intend to use this in production code? All of the ways I can think of implementing this involve introducing either unnecessarily complicated code or significant potential for spooky action-at-a-distance bugs. Both of these result in maintenance and code-reuse headaches.

Caveats aside, one fairly straightforward way to do this would be to set either a module variable or global variable in a signal handler so you can test in DESTROY if die was called. See die and %SIG for information on setting $SIG{__DIE__}.