I almost guarantee it will have a check engine light and it will be for Cat efficiency errors and O2 sensor errors.
Yeah, the reason why the fuel miles goes down is it is all designed to run as clean as possible and when something goes goofy in the cat O2 system it defaults to a rich concision so it doesn't damage the engine by going lean. Some vehicles, in fact most vehicle do perform better rich but you do loose fuel mileage. Some vehicles just run poorly with bad or disconnected cats and O2s and you would need a ECU or chip to correct for the lack of cats and O2s.
So, yes, to answer the question, you will have a light and it won't run properly if you do this without additional modifications to at least the ECU.