We can't know which ones, if any, are mandatory until we get our hands on the new content. It could depend on the healer's skillset and the encounter they are in. I believe when swapped, even inside a dungeon, there is no cooldown penalty. I am not that affected by this because I was already doing it with current tier spells such as stone 1/2 and ruin/broil in order to save a slot on my crossbar. We have to take into consideration that previous mandatory abilities may no longer be so.

Here are my thoughts:

Cleric Stance: Until brought up to 10% damage increase, this is collecting dust. 5% + 90second CD? It is nearly worthless

Break: Replaces Blizzard 2 if you want it. It has the same potency and does not scale down according to number of mobs. WHM doesn't need this at all since they already have so much burst AoE. SCHs might want to really consider bringing this along, and I can't speak for AST since I don't play it.

Protect: I believe this is no longer mandatory. Only the meta makes it so now. So little healing is required in 3.0 that you really don't need this outside of EX/savage. The only reason why I use it now is so I don't slow down the run with players demanding for it, and also because there is nothing to better to replace it. For WHM, since we need to figure out how to cast cure 1 and 2 without overhealing, you may want to ditch this completely.

Esuna: There are few encounters I know of that actually need this to survive it. Scorpion boss in Sastasha hard is one that comes to mind. It almost serves the sole purpose of helping to fill the LB gauge. We will have to keep our eye out on how often detrimental effects that can't be ignored are being given to you and your party members.

Lucid Dreaming: Almost seems meant for RDM. SCH may have had their Aetherflow cut in half, but I believe they are going to be ok. They literally would never have to worry about MP unless raising multiple players. Their potencies has been increased across on many abilities which I believe will help balance this out. For WHM, with Assize being reduced to 60 seconds, and the addition of thin air, they may need this more for the enmity reduction than the MP refresh. Again, we have to wait and see.

Swiftcast: Still mandatory. It is just too good in clutch situations.

Eye4eye: Great for trash mobs, sucks for bosses. Especially magic based ones, which are many. SCHs still need it for deployment tactics in the situations they feel it will benefit. My WHM will very likely be leaving this one behind, again due to the fact that mitigating damage is no longer their game, at least not through direct means.

Largesse: Again, WHM may no longer need this, but it was the one ability that I felt they truly lost as a job specific ability. I'm guessing this will only be needed for EX/savage content for all three healers.

Surecast: This went from zero to hero from what I can tell. I get stunned, A LOT. I get knocked back, A LOT, and I can think of a handful of bosses with draw-in effects. I don't think this is mandatory, but it has guaranteed a spot on my crossbar. I am really excited about this one.

Rescue: This deserves its own thread after we've have seen the tooltip and know that it does not decrease enmity. All it does is draw a player to your side. It kind of defeats the purpose of the name "Rescue", but I believe the best time to use this is after a player has been raised to immediately bring them in your range. I think we need to consult with DPS mains to let us know when, if any time they would like this done to them. Until we learn more, this too will collect dust.

tl:dr? The only mandatory role skill I see for all three healers is swiftcast.