Die from laughter
Die of laughter
Collapse with remorse
Collapse from remorse
^ All of these are OK and the first of both pairs means almost exactly the same as the second.
make haste out of necessity
^ This is awkward, though it would be understood. Perhaps try instead:
make haste as needed
Finally,
make haste from necessity
is completely incorrect use of the preposition 'from'. When "make" and "from" colocate, the indication is that the object that appears before "from" is being produced from the object after it; the second thing is being turned into the first.
Make bread from flour.
Make wine from grapes.
Make a mountain out of a molehill.