Commonest complication of Zenker's diverticulum is
Dysphonia
Gastroesophageal reflux
Lung abscess
Perforation
"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