If I ride a horse onto an active Nether Portal and dismount, the horse spawns in the other world, but it is partially inside the obsidian frame of the portal and immediately starts taking damage.
If you wait before entering the Portal yourself, the horse dies.
I have been able to enter the portal myself and quickly mount the horse on the other side and then move it away, but it (and the player) both take damage from being inside the obsidian frame.
This was not a problem in previous versions as I have always been able to move horses through the portals with no issues.