The My Best Friend's Wedding actor is currently starring as Professor Higgins in Bernard Shaw's Pygmalion in London's West End, and he admits he's turned to alcohol to calm his nerves.
And the Brit is convinced he will transform into a big drinker like Burton due to the stress of showbusiness.
He tells Britain's Daily Mail, "Apart from odd nights I don't stay up very late. The only thing that's going wrong is that I think I'm becoming alcoholic.
"I'm drinking more and rehearsing this play has almost driven me to alcohol because its so nerve-racking. Cider has become my new tipple!
"I can see myself ending up like Richard Burton, because the business is very stressful and I even started smoking again during rehearsals for this."
But the 51-year-old star is adamant he's too old to consume any harder substances like he did in his youth.
He adds, "You can't take drugs when you're 50. I want my brain to continue functioning. The thing I'm worried about most now is brain disease. Having taken so many drugs in my life, I think, oh my God, what about all my neuro things?
"I don't think I ever had a drug problem - I just took a lot of drugs. But slowly I stopped and if I take any drugs now, it's just prescription pills."