My professor always insists that I provide power to an oscilloscope thorough an isolation transformer. What is the necessity of this? What is the risk if I don't connect it?

You should never float a scope with an isolation transformer! This is reckless and dangerous advice from your professor, and he/she needs a reality check.

The accepted procedure for doing work that requires isolation is to ISOLATE THE UNIT UNDER TEST, NOT THE TEST EQUIPMENT.

Why? 


It's much easier to remember that the unit under test is what's unsafe and needs cautious handling, not your oscilloscope
If you hook a communication cable up to your floating scope (USB, GPIB, RS232), guess what - it's NO LONGER FLOATING. (All of these cables have earth-referenced returns)
As soon as you connect that floating scope return to a potential, all of the exposed metal on the scope is now at that potential. Major shock hazard.


If you cannot float the unit under test, use an isolated differential probe to do your measurements, and keep both the UUT and scope earthed. No measurement is worth the safety risk.

A battery-operated scope may seem like a good idea in this circumstance, but only if it has dedicated isolated inputs. A battery-operated ordinary scope with non-isolated inputs will still suffer the problem of the exposed metal floating up to whatever potential you connect the ground to. That's why all of the manuals for the battery operated scopes clearly say "This scope must always be earthed, even if you're running off the battery" - if you choose to ignore this, it's at your own risk. A scope with dedicated isolated inputs should still be earthed as a good practice. It's essentially the equivalent of using external isolated differential probes with an ordinary scope.

I work full-time in power electronics and have tens of thousands of dollars of lab equipment at my bench. If anyone is caught floating their scope, the float is immediately corrected by the test engineering team, the means of float is seized (most often this is a line cord with the ground prong removed) - disciplinary action is a possibility. Numerous senior/principal engineers have fried their PCs and their entire set of GPIB-connected bench instruments by trying to float the test equipment and forgetting about the GPIB interface. (No one has died yet - thankfully)
