While one is tempted to go with your description of "unburnt gas. It's not a hard gas smell, but it's not a full exhaust smell "...and head down the usual diagnostic path...I would check for obvious gas leaks FIRST in the name of safety. To start with, check under the car for drips, pop hood and turn ignition to on but DON'T START engine...then inspect engine bay for leaks with system pressurized...including injectors, rails, Schrader valve, fuel line connection onto flex hose to manifold, etc. Then look under left rear for drips from fuel filter. After proving me wrong here, shut off ignition. There have been a few filler hoses that partially separated from tank as well but that will take more checking than possible at home.
While others may disagree, I would not start this car inside your garage without a fire extinguisher handy (yes, I keep one in each Viper).
And now...onward toward your suggested diagnosis...your Viper is likely running rich due to bad plugs, old wires, failing O2 sensors, stuck
injector, etc. You may be dealing with a bad cat as well. Be sure that you didn't miss a check engine light (also be sure it works on start-up). If you have a code reader, hook it up to see if anything shows up yet.
Sounds like a check-up by a Dodge dealer with proper scan tools would be in order.
Sorry that I can't help further...but I am sure others will add/subtract to/from the above points.