I had a big creature (something like 10/14), and my opponent had a 4/1 death touch creature. I played Ruunin’s shrine and attacked my big creature into the death touch. As expected, the shrine took the damage instead of the creature, but neither of them were killed by death’s touch. The wording on the shrine seems to indicate that the damage is received by the shrine instead, which should trigger death touch since the damage still originated from the opponent’s creature.
Searching for this found an old reddit thread where the big creature ended up dying, so the mechanics must have been changed at some point.