I have a 2013 Limited where the driver seat warmer turns on upon engine start as well. Took it to the dealership and had the electrical guru take a look at it and he said that in 99.9% of these cases, it's the switch failing due to spilling something on them that was in the cup holder adjacent to it.
They were going to charge me $86.38 for a replacement and another $260 for installation. I found the part at
https://www.toyotapartsdeal.com/ for about $15 less and the install took about 5 minutes.
If that solves anyone's problems, awesome! It didn't fix mine so I'm still troubleshooting.
