I'll back up others on the Healer's telling tanks they need mana... it's not needed.
Tanks should be paying attention to what all is going on. Which means their healers mana, position of dps, anyone pulling aggro, their own health to use CD's effectively, etc...

Just like tanks would appreciate if healers paid attention to the tanks mana and tp, and NOT tell us to continue pulling when we're low on either and it's our main source to hold threat.

In the flip side to this, dps should also be watching for opportunities to use anything at their disposal to help the healer and tank. An idea of this is when I'm on MCH in a low level dungeon, two mobs are beating on an undergeared tank and the healer's low on mana. I'll use my knockback, then bind it in place to give the healer a moment where the tank's only being hit by one.
Or, I'm running as SMN with a AST, I'll then use E4AE and Virus, as I know the AST doesn't have access to them. And in 8 men runs as SMN, I'll tell healers to let me use my SC-rez first as I can regen my mana fairly faster than most healers.
Not to mention, dps should be aware that nearly every dps class has survival CDs. Tanks aren't the only ones who should be using these to help stay alive!