What could cause my burps to leave the taste of blood in my mouth?

Depends on food: burps are a release of gas from the stomach (or lower) through the esophagus and out the mouth. Most of this gas is from the air swallowed while eating or incorporated into the food. It could also be the gasses released from a carbonated drink (CO2). Assuming you are not having GERD or a gastric ulcer the taste is most likely from the foods eaten. May be an altered taste from the stomach acid.

Gingivitis: Bleeding taste can be a sign that you already have, gum disease. The bacteria and other particles, constantly form a sticky, colorless toxic “plaque” on teeth causing the gums to become inflamed and to easily bleed. This is a serious problem, which should be checked professionally by a dentist. The good news is that this condition is curable. Call your dentist today!