Are there English equivalents to Japanese word, ‘有名税－Tax on the famous’?

We say "the price of fame".

In an article with that title in Psychology Today, 9/1/1998, Raj Persaud wrote:

The constant attention that comes with fame inflates some celebrities' egos. For others though, the effect is the reverse: it makes them so aware of their shortcomings that they may be driven to self-destruction.

Mark Schaller, Ph.D., of the University of British Columbia, has surveyed the works of songwriters Kurt Cobain and Cole Porter and of writer John Cheever to see how often they used the first person singular. With each man, the rate of self-reference jumped after he became famous.

Schaller theorizes that the relentless scrutiny of fans and the media leads some celebrities to become acutely self-conscious. Some develop "impostor syndrome," he observes. "They think to themselves, 'I know that I'm not as great as they think I am.'"

The need to escape this agonizing self-awareness may lead some famous people into alcoholism, drug abuse, or compulsive sexuality, says Schaller. Porter and Cheever were both alcoholics. Using journals and letters, Schaller has found that Cheever's battles with alcohol apparently followed the periods of his greatest renown.

As for Cobain, the leader of Nirvana was addicted to heroin and ultimately killed himself with a shotgun. "Suicide has been called 'the ultimate escape from self-awareness,'" Schaller notes.

The phrase goes back to at least 1743, when "a Student of Oxford" (the Oxford Professor of Poetry Robert Lowth, a notable scholar of Hebrew literature, and author of the most influential English grammar text of the 18th-century) wrote

Who seeks [honor] must the mighty cost sustain,
And pay the price of Fame; labor, and care and pain.

Fairly commonly used is 'The price of fame.'

You can talk about the curse of fame

you might also talk about:

the merciless public eye
or that's what you get for being in the spotlight