Believe it or not they need very little maintenance. But there is a very important minor maintenance that will make all the difference. The ear tips and the binural (the stainless steel tubing that connects the ear tips to rubber tubing) can get small amounts of ear wax building up. Ear wax will eventually effect the acoustics of your scope. There is an easy solution.
Every few months take a pipe cleaner and dip it rubbing alcohol and clear the ear tips and binural.
For the scopes that have gone decades without cleaning, you may need to use compressed air to get the tubes clear.
A few minutes of work will keep your Littmann stethoscope working like the day you bought it.