I see two different types of injury to the leaves. The little tiny holes that don't go through all the way through could be caused by by Elm Leaf Miners. They seldom reach populations where control is needed, but I have seen them get pretty bad.
The holes that are chewed all the way through could be Japanese Beetles, if you have them in your area. This is the time of year when you would expect to see the damage from them.
If the leaves shown are typical of those damaged, I would not be at all concerned. It looks like less than 5 or 10% of the leaf surface is gone. If it is much worse, then we need to explore whether or not this is really a Chinese Elm, and not a Siberian Elm, or something else.
I mention this because Siberian Elms will actually hold up better if they lose a good percentage of foliage each year to insects. This slows growth and lightens the canopy load. This tree is so weak wooded, that it is just asking for trouble to try to achieve a lush full canopy. Michael A. Dirr calls this "One of, if not, the world's worst trees."