Commonest complication of Zenker's diverticulum is
"Pulmonary infection is the most frequent serious complication, and many patients are first seen after experiencing repeated episodes of pneumonia."-CSDT

Aspiration pneumonia & Lung abscess are the most frequent complications of Zenker's diverticulum.
Other rare complications are:
Perforation
Bleeding
Carcinoma
Dysphonia
Gastroesophageal reflux
Lung abscess
Perforation