If someone constantly washed their hands in uncomfortably hot (but not very painful) water for 10 minutes at a time, several times a day, could this end up “cooking” their nerves and finger/hand tissue over the long run (i.e., thermal-caused damage?)
Doubtful
No.
For most people, uncomfortably hot is going to be somewhere around 42 to 45 deg C. Tissue damage doesn’t start to occur until you get a bit over 50 deg C.
I know these numbers off the top of my head due to the fact that about 20 years ago I designed a bunch of equipment for a neurobiologist so that he could study the human response to hot and cold induced pain. When newbies first started playing with the equipment their heat pain threshold tended to be somewhere around 42 deg C, but after playing with the equipment for a few weeks they tended to get a tolerance to it and their pain threshold increased to about 45 deg C. The safety cutoff on the machines was 50 deg C, which I know worked because I accidentally triggered it a few times during development of the machines.
I’m not good at gauging temperature by touch, but 45 and 50 Celsius doesn’t seem like it’s a big sensation difference is it?
There are two pain receptors in your skin, TRPV1 and TRPV2. TRPV1 kicks on at about 43 to 45 deg C, and TRPV2 kicks on at about 50 to 52 deg C.
So once you get over the TRPV1 pain threshold, it just hurts. You are going to have a very difficult time telling the difference between say 45 deg C and 49 deg C.
And actually, humans are extremely poor at sensing absolute temperatures. We do better at sensing relative temperature changes. The classic example of this is to put one hand in hot water (not so hot that you burn yourself) then another hand in cold water, and let both of them get used to the temperatures. Then run both hands under the same medium temperature water. Even though both hands are exposed to the exact same temperature, one will feel warm and the other will feel cool.
So you’re not alone at being poor at judging absolute temperature. It’s called being human. 
ETA: Incidentally, TRPV1 is also activated by capsaicin, which is why hot peppers feel “hot”.
Is it possible to build up the TRPV1 tolerance to real heat just through capsaicin exposure? For example, if I were to handle chiles on a regular basis and get used to it, would the nerves in my hands also be better able to handle the pain from hot water?
Ask any chef. They will handle hot food and utensils all the time and become pretty tolerant of quite high temperatures.
Is that the same as capsaicin tolerance? Capsaicin is the ingredient in chiles that make them “hot”. I could see how exposure to hot temperatures could make you tolerate hot temperatures, but I’m wondering if tolerance to chile “heat” also makes you tolerant to temperature heat.
Since it’s a chemical activation rather than a heat activation it might be different. I dunno.