An internal combustion engine needs three things to operate: gasoline and air (compressed), and a spark. Optimizing the combustion is the tricky part. The spark must occur at exactly the right moment, and the gasoline/air mixture must be in the correct proportions. The ignition system takes care of the spark and its timing, and the carburetor supplies the volume and mixture of the gasoline/oxygen.
But there is one other very important aspect of combustion: cooling. Although the exterior of the engine will be cooled by either air or water, the inside also needs cooling, and this is particularly true with a 2-stroke. 2-strokes create more heat as they burn the gasoline/air--twice as much as a 4-stroke.
Cooling From Gasoline
The 2-stroke engine is cooled on the inside by the fresh (cold) charge of gasoline/air as it enters the cylinder--the principal cooling effect coming from the gasoline. The amount of gasoline entering the engine is controlled predominantly by the main jet. The size of the hole in the main jet restricts the amount of gasoline flowing. The bigger the hole, the more gasoline will pass through it.
More Oil Equals Less Gasoline
Increasing the jet size increases the flow rate, and vice versa. However, on 2-strokes, the amount of gasoline flowing will vary with the amount of oil added at the pre-mix stage (the oil and gasoline both must pass through the main jet). More oil passing through equates to less gasoline, which equals less cooling.
For instance, an engine running cool with say a pre-mix ratio of 32:1 will run hotter (greater chance of seizure) if more oil is added, by changing to 20:1. Therefore, it is correct to say that adding more oil to a pre-mix fueled 2-stroke will make the engine run in a leaner condition.
No Extra Oil for Posi-Lubrication Systems
Under no circumstances should pre-mix oil be added to the fuel tank of a positive lubrication system 2-stroke. For the reasons stated above, adding oil in this way will make the engine run hotter (leaner), which can lead to seizures.