When the Library Telescope is not in use or being moved the lens cap and tube cover should be firmly in place. This will keep anything, but particularly dust, from falling onto the eyepiece lens or the mirrors inside the tube. Additionally, the focuser should be moved all the way into the telescope when it is not in use or is being moved. This will make it less likely to be bumped against anything. When storing the telescope, the tube should be positioned with the mirror on the bottom and the eyepiece at the top. 
Generally, the following should be checked regularly: Check the viewfinder and make sure the battery is still working. Be sure to turn it off. This is the biggest problem with the Library Telescope that patrons forget to turn off the viewfinder draining the battery. The library should have extra viewfinder batteries on hand to replace as needed. They are not very expensive. Check that the viewfinder is correctly aligned with the telescope. Often patrons will turn the alignment screws on the viewfinder causing it to be misaligned. Check that all the pieces are present and secure.
Ensure that the contents of the belt pack are present: the instructions, the constellation guide, and the red light. Check the viewfinder and make sure the battery is still working. Be sure it is turned off. This is the biggest problem with the Library Telescope that patrons forget to turn off the viewfinder draining the battery. The library should have extra viewfinder batteries on hand to replace as needed. They are not very expensive. (Most, but not all, LTP scopes use a battery pack of AA batteries. These are more forgiving than the button batteries.) Check that the viewfinder is correctly aligned with the telescope. Often patrons will turn the alignment screws on the viewfinder causing it to be misaligned. (The instructions explain how the user can adjust the red dot finder if needed.) Visually inspect the telescope for possible damage: cracked laminate, dents in tube, loose parts, etc. 
Sometimes the viewfinder may be damaged and need to be replaced. Additionally, the focuser may need repairs. The telescope has very few moving parts. The most common problems are due to people moving the focuser of eyepiece zoom beyond their limits.These issues are usually not hard to repair. Pieces may be loosened and lost requiring replacement. Most of the time, these issues are due to children, or people acting like children, fiddling with things. Suggesting adult supervision is recommended. Most things one might fiddle with have been secured in some way.
Generally, it would be best if a person knowledgeable about telescopes was the one doing the necessary maintenance. This may not be possible so library staff could learn to do much of the maintenance, but, generally, patrons should be discouraged from trying to do maintenance.
You don’t!  Merely touching the mirrors and lens surfaces may cause damage. The optical surfaces have special coatings on them that can be damaged if cleaned too often or to aggressively. It takes a remarkable amount of dust to effect the performance of the telescope. (Try covering half the opening of the telescope with your hand. It will only darken the view a bit.) If you think the mirrors need cleaned the library should contact their local astronomy representative for evaluation.  The eye piece is sometimes subjected to probing fingers, particularly very young ones, and so collect smudges. A very gentle cleaning with a tissue moistened with a little Windex should be enough.
Again, don’t. Many telescopes that have been in constant use for years have not needed adjustment. When, for some reason, adjustments are needed, there are simple methods that do not need tools other than to move or gain access to the 3 adjustment knobs behind the main mirror. Ask for help by contacting us . A call to your local astronomy club will likely get you someone to precisely collimate the mirrors, using a laser. The need to make these adjustments is usually due to someone trying to do it at home themselves.