Rhys said he was returning to Wales for a series of shows across the country, ending with a special performance in Bethel Chapel Café in Burton's birthplace Pontrhydyfen in Neath Port Talbot.
He said: "Michael Sheen has been putting all us Welsh actors to shame by setting up Welsh National Theatre so, when a space appeared in my schedule, I felt it was my time to step up.
"When Covid hit, I prepared a production of Playing Burton but have never had the chance to perform it on stage.
"With 2025 being both Burton's centenary year and the first year of the Welsh National Theatre, it felt right to return to the Welsh stage for the first time in 20-plus years," he said.
"The reason I wanted to act was because of Richard Burton.
"Since first seeing his incredible performance in Look Back in Anger to still listening to his audio of Hamlet and Under Milk Wood. He blazed the trail for us all and showed us it was possible."
The one-man show, Playing Burton, looks at Burton's beginnings in south Wales to his rise as one of the greatest performers of his generation.
Charting his passionate love affair with Elizabeth Taylor and his very public struggle with alcoholism, the play explores fame, ambition, and identity.
It was written by Mark Jenkins, directed by Tony Award-winning director Bartlett Sher and produced by Maris Lyons.