1. Does Glycerine burn?
Yes. Flash point 160C Autoignition 370C
2. Does it turn into toxic substances while burning or heated?
Yes. Emits toxic fumes under fire conditions. Probably CO and CO
2 would be my guess.
3. Does it do any harm to the skin?
Could cause skin irritation. Wash off immediately with soap and water.
As with any chemical check the MSDS before you use it and take appropriate safety measures, for example: safety goggles, gloves, appropriate clothing, respirator (if required), adequate ventilation, correct fire fighting measures, shower, eye wash.
Even though glycerol is not classified as hazardous this doesn't mean you can't hurt yourself or others with it. Check the MSDS and be safe.