Will there by any copyright issue if we decide to add ACM Computing Classification System to our theses? Let us say that we decide to use ACM Computing Classification System in the undergraduate and graduate theses from our department. These theses belong to the field of Computer Science and Engineering. 

Will there by any copyright issues from ACM side?

There is an ACM Chapter in our department.
Copyright applies to the text of the classification, not to the use of it. Giving the classification of a document would be roughly like citing a result in another paper (and is the purpose of the classification). There would only be a copyright issue if you want to copy out part of the text.