I had exactly the same problem: burning oil or exhaust smell from the moment I took delivery of my 2012 Fiat 500 on Sept 30/11 until one week ago. The burning smell appeared after my car had warmed up and when it was idling at a stop. The dealership refused to acknowledge that there was a problem because they could not replicate it. Chrysler-Fiat refused to acknowledge the problem as well and just suggested that I take it to another dealership. The worst customer experience I have ever had with any product I have ever purchased - from both the dealership, Downsview Chrysler in Toronto, Canada and Chrysler itself.
I managed to get the problem fixed through no help from Downsview Chrysler or Chrysler-Fiat, but through dogged persistence in not letting Downsview Chrysler off the hook. I drove the car for 9 months and kept on checking my car in for repair of the burning smell, each time getting brushed off until last week I dropped the car off and said I didn't want it back until it was fixed.
They found an oil leak which required the replacement of five upper exhaust manifolds to the cylinder head studs (reference on the invoice was "Performed Star Case S1109000025"). Fortunately this major repair was covered by the warranty.
The problem has been eliminated and I have been driving my car for the past week with air from the HAVC system that smells like roses. What a pleasure. But my next car will not be a Fiat because of the way I was treated.
Truly unacceptable service for a brand trying to make a comeback. VW pulled the same garbage when they "returned" or revived themselves. They treated people like crap, refused service, had some clear and defined issues they did no acknowledge....who are they trying to fool???
Given fragile Fiat's position is in this country and elsewhere, Fiat USA must be out of their minds to not address these issues quickly and professionally. I mean think about it...what an incredibly big deal they made out of helping you and look where it got them....you will never buy another Fiat again most likely.
Sad.
This is the major worry I have with this car....moreso than anything else. Many of my friends told me I'm out of my mind buying this car. We'll see if they are proven right or wrong. I don't want to spend a SINGLE minute at a dealership if I don't have to.
Sometimes I truly envy people who don't need to drive or need a car. Lucky them.