Death of a Salesman, the "greatest American play", is also extremely popular in, of all countries, China. No, not because it's a criticism of capitalism and the American Dream, but because of its intense focus on father-son relationships.
The Crucible is also popular, especially with the the generation that lived through the Cultural Revolution. Arthur Miller wrote that he had learned to expect to hear that it was being performed in countries where either a dangerously authoritarian administration seemed to be on its way, or one had just been overthrown. He was accused of being a Dirty Commie in the 60's.
Les Misérables is wildly popular in Japan, with a total of six Japanese-language cast recordings and has been running in Tokyo for over 20 years. Lampshaded in the game Final Fantasy VII, where posters of the play 'Loveless' that resemble the iconic Cosette poster are part of the dour backdrops of Midgar. There is even a doujin fighting game based on Les Miserables.
Several big spectacle musicals like Les Mis have had stagings in other countries and have become very popular in some of them. The Phantom of the Opera. In Hungary and Poland, the versions are not "clones" of the original English version, as in they have all-new set and costume design and other elements while retaining the songs and story-they have become popular enough that the creative teams are given new free rein. The Lion King had its tenth anniversary celebration in Johannesburg in 2007, and ran for about a year there.
Fiddler on the Roof is also extremely popular in Japan, due to the resonance of its themes about the decline of tradition and family.
The productions of Cirque du Soleil are much-loved in its home country of Canada, and they have traveled well worldwide (due in part to being on the visual side). That said, Japan deserves special mention. Dating back to their arrival with the country-specific tour Fascination in 1992 (an arena-based compilation of acts for Cirque's previous shows Le Cirque Reinvente and Nouvelle Experience), their first outside of North America to succeed, it's proven extremely popular there. In 2008, it became only the second country outside of the U.S. to get a permanent Cirque show, ZED, at the Tokyo Disneyland resort (it closed at the end of 2011, but that was due to a tourism slowdown post-earthquake/tsunami). This affection has even reached anime via the Kaleido Stage.
Starlight Express was probably Andrew Lloyd Webber's least impressive work, yet is still running today in Germany. They specially built a theatre just for Starlight Express in Bochum, including race tracks on three levels on stage as well as two looping through the audience. It would be really weird to stage anything else there. Panorama can be viewed here. (Requires Java, "Hohe Auflösung" = high resolution, "Niedrige Auflösung" = low resolution.)
In his notes to Heartbreak House, George Bernard Shaw noted that at the time (i. e. before and during World War One), Germans took their Shakespeare more serious than the British. The relevant section is headlined: "Unser Shakespeare".
In-universe example: a play written by the protagonist of the Israeli sitcom Life Ain’t All and stolen from him (he didn’t care for it much) became very successful in Hungary.
Dinner for One is a British vaudeville which is obscure at best in the United Kingdom where it hasn't been shown on TV even once but a Cult Classic in many European countries, starting with Germany where it has been aired by just about every PBS station each New Year's Eve since 1972.
Grease is, as well as its soundtrack, immensely popular in Sweden, within the Greaser and Rockabilly culture.
Our American Cousin has Lord Dundreary, who is this and an Ensemble Darkhorse. His popularity in America was due to him being an exaggerated British stereotype in a play that was mocking Americans.
La muette de Portici has somewhat of a following in its native France as it was the Trope Namer of Grand Opera, but is hardly well-known there outside of opera enthusiasts. In Belgium it is one of the most recognizable opera's ever, as it is taught in history classes as being one of the reasons why Belgians wanted to become independent.