Ad-hoc musings with no settled knowledge of the answer:
The first part of the answer is easy enough to find out. Get yourself a cup of tea and drink it while it’s slightly too hot. Follow the burning sensation down to where you no longer feel it. That’s where there are no nerve endings any more.
Should be at the bottom of your oesophagus, I think. Or maybe halfway.
Whether there are any nerve endings between there and the exit, I don’t know. I don’t think there are any. Your stomach is full of acid that you only feel burning when some of it comes up with a belch or when you throw up.
Eating very spicy food can also show what parts have feeling. It burns when you eat it and it burns on the way out, but nowhere in between.
Wait a second. Spicy food doesn’t burn in your oesophagus either, contrary to hot tea. Maybe temperature sensation uses different kinds of nerve endings?