Bile has a pH of 7.6 - 8.6 and is never, in itself, the reason to feel a burning sensations.
And bile never enters the stomach directly from the gallbladder. It goes directly from the gallbladder into the duodenum (which is right below the stomach). However, if the bile (for some reason or another, I believe vomiting is one) would happen to go backwards and up into the stomach, the stomach would react by creating more stomach acid to compensate for the immediate influx of alkaline bile (the stomach NEEDS to stay acidic for digestion).
But normally, the chewed food first gets the 'stomach acid treatment', being churned and mixed with the stomach acid for that part of the digestion process; then it goes into the duodenom/upper smaller intestines, where it's mixed with bile from the liver or gallbladder...and then it gets the 'bile treatment' for the further digestions and assimilation, and the breakdown/utilization of fats. Consuming fat (or oil swishing) triggers the production and release of bile...but with "systems normal" this should not cause a burning sensation.