Are the odours added to natural gas completely flammable?
On looking it up on Google and a few more places, it turns out that natural gas is primarily methane, along with a few other hydrocarbons.
Well, I have found out through some searches online that in the case of propane based fuel, like LPG (Liquid Petroleum Gas) in India, ETHYL MERCAPTAN is a popular option. It's smell is described as a 


  strongly disagreeable odor that humans can detect in minute
  concentrations


And that it's smell resembles that of


  leeks, onions, durian or cooked cabbage, but is quite distinct.


(source :Wikipedia)

And also, most important point of all, it is highly explosive.

This is what the New Jersey Department of health and safety has to say:



(Source : This document)

So yeah, You're right. Based on this example, I think it logically follows that these "additives" are flamable and result in products that are ODOURLESS.