Hey long time since I 3-healed this boss so hope I can pitch in with anything.
First of all to deal with transitioning to phase 3 I totally agree with Myth, if you have the option of prolonging phase 2 in return of going into the nether without a deathcaller it will ease entering p3 a bit. Should you be in a position where you go in and you have the Deathcaller with you, just run to either of the sides and get close enough to heal tanks. Like this http://imgur.com/trcZlD1
Upon closer look on the logs it seems as though the tank tanking the Deathcaller also tanked Archimonde in p3, he went from 100% to dead in 1,7 seconds. In my experience only the tanks personal or absorbs is able to save him from something like this. Make sure that he isn’t tanking the boss AND the Deathcaller at the same time, it will guarantee these combos, might as well have the ‘free’ tank on Archimonde until the situation has been stabilized. Looks like tanking is the issue, not so much healing (I can see on the footage how fast he dies). Hadn’t he tanked Archimonde while tanking Deathcaller then he wouldn’t have died. Link to the point in the log: https://www.warcraftlogs.com/reports/gWdPqjbJrBZ8znkx/#type=deaths&death=1
On this attempt though he died only to the Deathcaller (over a longer period), and it seems like it’s just healing being a bit behind. You were a healer short here and holy paladins have great tank healing while they’re out of range (He died to getting double-tapped by wrought chaos): https://www.warcraftlogs.com/reports/gWdPqjbJrBZ8znkx/#type=deaths&fight=2&death=3
Just eliminate stupid deaths and don’t let tanks hold both Archimonde and Deathcaller in phase 3.
Hope this shed some light on tanks dying!