I am pretty sure that nothing would happen. I am presuming you are referring to a vampire's create spawn ability that triggers if a creature is slain by their energy drain or blood drain ability (which happens when they bite a creature).
I assume that if the clone is inert, that means the original template creature hasn't died yet, so it has no soul in it. It would likely have the same hit points as the original body (at least, based on the time the flesh sample was collected, in case some time had passed).
I would say that an inert clone is more or less just like a slab of meat, like casting stone to flesh on a boulder (although the clone is more than just flesh, it would be bone, blood, muscle, brain matter, etc).
So I'd say a vampire could drink the blood from the clone, and if the Constitution loss from that would reduce the clone to 0, the clone would 'die', though just basically meaning it's ruined and wouldn't work if the original died. Its soul wouldn't go into the clone.
Since vampires are intelligent undead, and they typically are creatures that had souls and its pretty determined that intelligent undead have lifeforces (just negative), I would likely rule that an inert clone couldn't become a vampire. It would just be a soulless hunk of meat that rots away (unless preserved). I would allow such a corpse from a inert clone to be animated as a skeleton or zombie or other suitable undead if it were 'killed'. Since animating mindless undead typically involves animating them with an elemental spirit or something, not the spirit of the original creature.
