Proton pump inhibitors are fairly innocuous drugs so should have no effect. Lots of people are on them.
With regards to Barrett's oesophagus, it is in fact a premalignant condition, ie left untreated it may progress to cancer of the oesophagus (although I stress that it is not cancer itself). The good news is that with appropriate treatment, ie PPIs, the condition is reversible and hopefully should never progress any further. You will need further surveillance in terms of endoscopy to keep an eye on it. But at the moment you are entirely healthy and it should have no effect on your licence.
Ps if you smoke - stop!