I did an experiment in my school lab in which we had to identify the given organic compound. In my Lab Manual, it's written:
Take 2 ml of Tollen's Reagent and add a few drops of organic sample. Heat the contents on boiling water bath for 5 minutes. Silver Mirror is formed. The given organic compound is Glycerol.
As far as I know, the compounds with aldehyde groups in them give silver mirror. Also those which contain $\alpha$,$\beta$-hydroxy ketone are known to react with Tollen's Reagent. I have never heard of Glycerol reacting with it. I searched but couldn't find this thread anywhere. Though I found that on heating Glycerol, it gives us Acrolein which does contain aldehyde group, which I think is the reason but I am not sure.