This patient most likely has hemoglobinuria, possibly due to intravascular hemolysis from his mechanical mitral valve, whose dysfunction is suggested by the finding of mitral regurgitation on physical examination. Fragmentation hemolysis in this setting manifests as a microangiopathic hemolytic anemia with thrombocytopenia and is accompanied by the release of free hemoglobin into the circulation. Free hemoglobin is partially bound by haptoglobin but may also be filtered into the urine, producing a red color. Heme reacts with peroxidase in the urine dipstick, causing a false-positive result for blood. Hemoglobinuria is distinguished from true hematuria by the absence of erythrocytes on urine microscopy. Similar findings on urinalysis will also occur with the release of myoglobin into the circulation, usually from muscle injury (rhabdomyolysis). Myoglobin is a small molecule relative to hemoglobin, is not bound within the circulation by haptoglobin, and is readily filtered through the kidneys, resulting in red-colored urine. It also reacts with peroxidase in the urine dipstick indicating blood, although microscopic examination will also be negative for erythrocytes.

Bladder cancer is a concern in a patient with a significant smoking history presenting with a finding of red urine. However, the urine color change in bladder cancer is due to bleeding into the urinary tract, and erythrocytes would be seen on urinalysis.

Glomerulonephritis may be associated with bleeding into the urine and would be suspected if erythrocytes, particularly acanthocytes (dysmorphic erythrocytes), were found on urine microscopic examination. Proteinuria may also be found in glomerulonephritis, although this patient's proteinuria is relatively mild and may result from tubular damage caused by hemoglobin toxicity.

Nephrolithiasis often presents with true hematuria in association with acute flank pain radiating to the ipsilateral groin, with or without costovertebral angle tenderness. Despite his history of nephrolithiasis, this patient does not have suggestive clinical symptoms and has no evidence of erythrocytes on urinalysis, making this an unlikely diagnosis.