Remember the Lazarus device can be used on both incapacited AND dead hunters. Always use it instead of the standard revive animation.
Keep your distance. Lazarus is a massive target for the monster, and if he's killed, things will go south quickly. 30 meters is all you need to heal a comrade, further out than that is even better. If the monster gets on you, pop your cloak and re-position to safety.
Lazarus' Sniper Rifle can be fired in rapid succession, but quickly loses accuracy. If you want tightly packed marks or headshots instead of a wide spread, fire slightly slower and aim down the scope.
Unlike other medics, you don't need to worry too much if your teammates take damage and die. Let the monster kill an ally then go after someone else. While its distracted and going elsewhere, cloak and revive the ally with your Glove.
Allies revived by the glove do not have any Health penalties, but have only 80% health. A glove shot and healing burst will bring them to full immediately.
The Lazarus Device has a decent arc, and can pick up fallen wildlife in addition to teammates. Be careful to aim as far away from Tyrants or other dead creatures as possible when using the glove on teammates.
Monsters will often try to eat the bodies of your slain teammates. Do not let that happen! Pester the monster with sniper rifle shots until it moves away, and call on your allies to help keep it from eating.
Be patient when going for the resurrection. Experienced monsters will set up camp close to the body to interrupt Laz's glove. To counter this, wait patiently for your moment and move in quickly when the monster is distracted or forced to retreat. Coordinated teams can do enough damage to make a camping monster pay dearly for a strike, making Lazarus effective without ever using his glove at all.