No doubt you are in a particularly difficult position, but I'm sure many readers who try to help others feel the same way; trying to get people to change can be very discouraging. People said of Jeremiah, "Let's attack him with our tongues and pay no attention to anything he says" (Jeremiah 18:18).
How can you avoid discouragement and burnout? First, be grateful for the victories you do see. Yes, perhaps many people you deal with won't change very much–but how much worse off would they be without you and your colleagues? And when you do see someone turn their life around, thank God for using you to help them.
Then take time to rest and get away from your job. Don't feel guilty–God made you, and He knows you need rest. In the midst of an overwhelming schedule, Jesus told His disciples, "Come with me by yourselves to a quiet place and get some rest" (Mark 6:31).
Finally, believe that God has called you to this position. You aren't there by accident; God put you there for a purpose. Every morning, commit your day and yourself to Him, asking Him for wisdom and strength. Remember: "Our help is in the name of the Lord" (Psalm 124:8).