I will highlight some examples I have seen.
A student is off task while using a computer.
This is actually a behavior problem. The computer has simply replaced another way to waste time - looking out the window, passing notes in class, sketching something meaningless in a book instead of listening, daydreaming while looking out the window, reading an unrelated book or magazine in class, etc.
The problem can be reduced by being alert. I was interviewing a job applicant recently and she mentioned that she simply teaches (at times) from the back of the room so that she can see the students' screens.
An experienced teacher can often tell just by body language and eye movement...
It is behavior management not technology management in many cases.
Marking a roll/recording absent students.
When we moved to an online roll marking system where a roll was marked in each lesson, it was noticed that some staff weren't marking a roll regularly. Guess what...this was nothing new. The same staff didn't mark a paper roll accurately and regularly. However, now that it is centralized and easy to access, this becomes obvious.
Students who don't access homework in an Online Learning Environment
Quite often these are the same students who were tardy with homework when it was delivered in the traditional format. However, a centralized system where it is easy to access this type of data makes it obvious.
Ensuring documentation is completed fully and correctly
When documents are in paper form and are filed in a filing cabinet 'somewhere', often accessed by only one or two people, it is difficult to ensure that all has been completed properly (until something happens, there are legal ramifications, and the flaws are exposed).
My school recently moved to an online system where parents approved their child's participation in an excursion via a centralized online system. This allowed administrators to easily see which parents had given permission before the excursion took place (with the possibility that a student had slipped onto the excursion accidentally - a potential legal minefield).
Initially, it took some extra effort to follow up some parents/students. This was seen by some as a new 'nuisance factor'. In reality, there had potentially been situations in the past where students had attended an excursion without permission and it wasn't noticed.
The technology didn't produce the problem, it merely allowed a problem to become more visible.
Suddenly we have students who are plagiarizing significant parts of assignments. No, some students have always done this. Now that it can be more readily discovered through integrating systems like TurnItIn into the Online Learning Environment (in Desire2Learn - the OLE that I use - it is a simple tick of a box in the DropBox assignment submission area) it is easy to identify - and reduce.
(BTW - this was not a significant problem at my school when this system was introduced, but I have heard of schools where it is a problem.)
No doubt there are similar examples in other industries...