While you shouldn't try to eliminate the warming sensation of bourbon, as it adds complexity to the spirit, there are ways to lessen the burn's dominance.
One simple way to do this is to add a few drops of water to a glass of bourbon. You'll notice a change in flavor, with some of the heat taking a back seat.
For further effect, serve the drink on the rocks; ice will significantly soften the hot sensation. Plus, melted water will mix into the bourbon, further diluting the spirit.
Finally, you can use bourbon in mixed drinks to eliminate the burn. Cocktails are typically chilled and diluted with other ingredients, reducing the alcohol content.