Mint Gum has metanol in it which is like a cooling agent and it tricks your nerves into thinking its something cold like an ice cube in your mouth. And then you drink the water, and your brain thinks its extra cold and it just feels like you swallowed alot of ice.
The flavouring substances in gum e.g. mint etc., cause localised vasodilation (the veins and capillaries in your throat dilate). The tonsils at the back of your throat in particular are highly vascular. When you drink icewater, this causes sudden and severe vasocostriction (a vasopressor response to cold immersion), which results in pain. You find the same effect if you drink something very cold after drinking something hot, and the same principle applies in 'ice-cream headaches'. It's the relative degree of change in vasopressor state that results in that feeling of burn or freshness.
