I have done a lot of tanking up to savage content, with plenty of dungeon runs, and I will happily say it is the tank's issue easily over 95% of the time. The job of the tank is to be the leader and team-player, working to the strengths and weaknesses of the particular healer they are with at the time.

The tank has full control of the pace of damage incoming on himself during dungeon runs, dictated by number of enemies pulled and cooldowns available... as well as paying careful attention to the gear, healing output and general level of ability of the healer. The very first thing I do as a tank in a dungeon is check to see how well the healer responds to damage on the first pack. If they seem a little undergeared or a bit slow to respond to the damage incoming, I know not to make door pulls.

Many MANY tanks in dungeon runs do not understand the concept of damage smoothing, and will mindlessly stack several cooldowns at once and take no damage for several seconds... then almost immediately die to raw unmitigated damage. Many tanks do not realize that it is absurd to use all cooldowns on the first door pull, then none on the second and expect to live.

Furthermore, if you're running OOM as a healer it is most likely because the packs are not dying fast enough and you're required to cast an excessive amount of holies. This can also be a DPS problem if your DPS are not putting out enough damage to clear large packs before you and the tank's cooldowns run dry. Sure you can cast holy, but even with WHM damage you can only contribute around a maximum of 30% of the damage dealt in a competent party.

Otherwise, on large pulls you do need to start hard healing when your chain of holy stuns runs out, generally speaking. You can holy a few times and obviously the tank will not take any significant damage during that time due to the stuns. After that, you will almost always need to start hardcast healing on very large pulls, unless the tank is (wrongly) stacking multiple cooldowns at the same time.