Being an anxious healer myself, the biggest piece of advice I can give you is to let him/her know it is not the end of the world if you die, or the party wipes. Wipes, tank deaths, and being yelled at tend to be the biggest fears. If your healer is overly worried about messing up, guess what they're going to do? Even to this day, if I enter an instance and the tank is being polite and considerate, it just melts a ton of that anxiety away.

If your healer announces that they are new to the instance, that is the perfect opportunity for you to take a minute to explain the boss fight mechanics to them when you get to them. Don't ask if they know the fight, they don't. They can read a guide and they are still not going to know the fight. Also, no ignoring mechanics.

Recognize that the action is going to be 10x faster for a new and/or anxious healer, which will increase the margin for their error. Because of this, you want to start your trash pulls easy and not pull multiple packs. Just because you don't normally run as healer doesn't mean you don't know how healers can make things tougher for you. If the new healer is spamming medica, fluid aura, casting cure 2 when only cure 1 is needed, it is ok to let them know these things. The healers that want to grow and get better will be receptive, especially if you were polite to them in the beginning. I promise.