Heat in barbecue sauces can come from a variety of sources, including hot chile peppers -- which can come fresh, dried, flaked or as a powder -- peppercorns and hot mustards. These substances can add lots of flavor, but it's easy to overdo it on the heat. This situation can typically be corrected, but the best method to use depends on the composition of your sauce.
Sugar is a sensible first option for bringing down the heat -- barbecue sauces often already feature sweet flavors, and some added sugar in these can tone down the sauce without dramatically or unfavorably altering the flavor of the sauce overall.
Add an Acidic Substance
Acidic substances can help counteract the burn. Tomatoes, which can be easily added pureed, crushed or as a tomato sauce to a prepared barbecue sauce, are an obvious choice for sauces that already include tomatoes as a main ingredient. After adding the tomatoes, retaste the sauce and make corrections as needed to the flavor to balance it out. With barbecues sauces like North Carolina-style vinegar-based sauces or South Carolina-style mustard-based sauces, acidic lemon juice or additional vinegar make sense to bring down the heat of the sauce.