This mixture, often called "chromic acid," is a common way to clean glassware. It is a strongly oxidizing acid which reacts with most organic material, so it is able to remove most types of organic sludge. You are likely smelling some mixture of the oxidation products.
It is very important to clean up Tollen's reagent, as it produces an explosive upon standing. More info here, including an accident report. However, chromates and dichromates are carcinogens and need to be disposed of as hazardous chromium waste. Making the spent (or unused) reagent safe only requires a dilute acid, so in most school labs, the spent Tollens reagent is disposed of with dilute acid and lots of water, producing fewer hazardous byproducts. This does not remove the silver mirror, which could be removed with nitric acid... or you can just dispose of the test tube. Chromic acid is overkill and most people would not risk exposure to chromium(VI) to save a test tube.
This paper  is very old, but it describes the reaction between silver, sulfuric acid, and potassium dichromate. I haven't looked beyond the first page (I don't want to pay for a paper from 1837!), but it tells you want to expect. Perhaps the ability to react with the silver mirror is why your teacher uses chromic acid despite the hassle and risks.
 R. Wazington, Esq., "On the action of chromic acid upon silver, and its combinations with the oxide of that metal," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1837 (first page).