Even though I prefer to do dps, I can get into groups a lot easier by healing. I'm currently level 40, so I don't know if this is a problem in veteran ranks.

First, players need to learn how healing works in this game. Most healing spells have a small circle of effect. If you are not in that circle, you can't get healed. Groups need to stay together so the healer isn't spending his/her entire time running around to different players healing them all individually and running out of magicka because of it.

For example, I just got done healing in the Blessed Crucible. There are some large areas in that dungeon. Each player was in different corners, separated by at least 75 yards. I had to stand in the middle of them spinning around pumping my restoration staff to get enough magicka to heal all of them. This is why groups get wiped.
We shouldn't have to use a healing potion every time it's ready because we're focusing our magicka in different corners of the room.

Second, if you aren't the tank, don't initiate the fights. It seems like every run there is a nightblade trying to get the first blow sneak attack on every group of mobs. Then that nightblade ends up having to run half way across the map trying to get away from the mobs because he/she can't take damage and now I have to chase them down to fix this. This is why I have to keep crowd control spells on my bar instead of buffs or more heals.
Third, if you are a tank, you aren't as damage proof as you think. The only reason you are alive is because you have constant heals on you. Don't go aggroing every mob in a room. It doesn't go much faster and you, and the dps, can't take that much damage.

Lastly, if you follow these tips, the group has a much lower chance of being wiped and the dungeon can go even faster because then your healer can focus some of his/her magicka on dps and take down the bosses quicker.

TL;DR Stay in a close proximity of your group members, don't start fights if you aren't the tank, don't aggro an entire room, and you will be more efficient in dungeons.