Acute Coronary Syndrome (ACS): This condition, which includes myocardial infarction (heart attack), is a leading cause of chest pain and shortness of breath. The "back burn" could be referred to as a radiating pain from the chest to the back, a common presentation in ACS.

Pulmonary Embolism (PE): A blood clot in the lungs can cause sudden onset of shortness of breath and chest pain, which might be described as a burning sensation.
Pneumonia: Infection in the lungs can lead to inflammation, which might cause chest pain and shortness of breath. The pain could be sharp or burning, especially when coughing.
Gastroesophageal Reflux Disease (GERD): Stomach acid flowing back into the esophagus can cause a burning chest pain and, in severe cases, difficulty breathing.
Costochondritis: Inflammation of the cartilage that connects the ribs to the breastbone can cause sharp pains in the chest that might be described as burning, especially with deep breathing.

Aortic Dissection: A tear in the aorta's inner layer can cause severe, tearing chest pain that radiates to the back. It's a medical emergency that requires immediate attention.
Pneumothorax: Air in the space between the lungs and chest cavity can cause the lung to collapse, leading to sudden and severe shortness of breath and chest pain.
Esophageal Rupture: A tear in the esophagus can cause severe chest pain and difficulty breathing, especially if it leads to infection or abscess formation.

Pericarditis: Inflammation of the sac surrounding the heart can cause sharp chest pain and shortness of breath. The pain can improve with leaning forward.
Myocarditis: Inflammation of the heart muscle itself can lead to chest pain, shortness of breath, and other symptoms similar to a heart attack.
Spinal or Thoracic Disk Herniation: Though less common, a herniated disk in the thoracic spine can cause severe back pain that radiates to the chest and can lead to breathing difficulties if it compresses nerves affecting respiratory function.