I can think of several things:
- If neuroscientists find that long term memory is encoded in something not preserved by cryonics. (Such as pH levels around certain neurons, or some weird quantum phenomena.)
- If small animals such as mice are trained to perform some task, then fail to demonstrate that training after being cryopreserved & revived. This requires technological advances, but not as much as human-level revival.
- If scientists discover a theory of consciousness that precludes its resumption after cryopreservation-like circumstances. (This would also mean that philosophical zombies are possible.)
Some of these may seem like a tall order, but if cryonics doesn't work, then that would tell us something new about biology, physics, and/or the nature of consciousness.
So far, we've seen all the evidence you'd expect to see if cryonics works: Memories are preserved even after cessation or disruption of electrical activity in the brain. Embryos and other tissues are cryopreserved and thawed without problems. Small mammal organs have been successfully cryopreserved and thawed, and remained functional. SEM micrographs of cryopreserved brain tissue show intact structure, including axons and synaptic vesicles. Most of the problems seem to be in the implementation, not the principle itself. (eg: avoiding warm ischemia when people die suddenly)