I would be raising turn 70% of the time or so in this spot. I would think the SB has JTxx possibly but not necessarily hearts, and I would be raising to get it in free rolling the AJT8 outs to bigger nuts.

The other 30% I would be calling to pick off river stabs and bluff out chops if it pairs and SB checks. If I were calling turn I would tank a little to rep decision.

If the heart hits river and they check I am unsure.... players will check here with mid flushes some of the time to allow for bluffs, and we have no hearts in our hands.... likely I would still bet if they checked and an unpairing heart hit.

100bb effective punt it in, would be surprised if villain folds to often and you don't want to lose value vs a set if the river comes a H for example

What should we think about his range with that line? Why would he play a set or JT that way?

There are a percentage of players that have a very high donking frequency. I have played against one reg in particular where I have modified my lines to be betting whenever they check, and calling the top of my range on the flop to raise most turns because their donking is unbalanced. They donk their moderate to very strong hands and generally check fold almost every other time. When I see donk bets like this where it is close to pot IMO it is often a naked JT that wants to get money in and does not want the flop to check through.