I figure that when the Heralds broke the oathpact (a very dishonorable act), Honor/Tanavast was somehow harmed by this. He presumably had Invested in them quite a bit, so perhaps this act (which was very against his Intent) somehow took enough of his power away as with Preservation binding Ruin and started him on the path to death. A couple thousand years later he was rambling on about humans destroying the planet during the Recreance and Tanavast died shortly after.
Perhaps Odium wasn't responsible for Tanavast's death (at least directly), but simply took advantage of that afterwards and splintered the Shard. The claim that he was killed by Odium at the end of the Way of Kings might have been more metaphorical (as the torturings on Braize by his people drove the Heralds to break the oathpact).