Because the board is not paired and there are only 3 hearts, the only hand combos that beat hero are 7h4h, 9h7h, and AhXh (where X can't be 5, 6, 8, J, or K). I doubt this villain isolates with 7h4h or 9h7h. Even Ah2h-Ah9h are questionable.

So villain wins with AhQh, AhTh, and maybe Ah9h, Ah7h, Ah4h, Ah3h, and Ah2h.

But what would he do this with that hero beats?
QhTh. Maybe Qh9h and Th9h. Maybe sets. Maybe some pure bluffs.

If I were the hero, at game speed I would certainly call river (I don't think I could ever fold a K-high flush on a non-paired, only 3-heart board). Taking the time to look at it, I guess it depends on whether villain would do this with anything worse than QhTh. It only takes a few more hand combos to make hero's call profitable.

Also, MadMojoMonkey, you don't think villain would widen his preflop range for the purposes of isolating a fish?