The original 'Game Boy' version of Tetris released in 1989, did not have battery backup.

However, the updated version for the Game Boy Color, Tetris DX (released in 1998/99), was backwards-compatible with the original Game Boy, and does feature battery backed high-scores:

developed by Nintendo and released in Japan on October 21, 1998, in North America on November 18, 1998, and in Europe and Australia in 1999. Tetris DX features battery-saved high scores and three player profiles...
_{Tetris (Game Boy) - Wikipedia}

The original 'Game Boy' version of Tetris released in 1989, did not have battery backup.

However, the updated version for the Game Boy Color, Tetris DX (released in 1998/99), is backwards-compatible with the original Game Boy, and does feature battery backed high-scores:

developed by Nintendo and released in Japan on October 21, 1998, in North America on November 18, 1998, and in Europe and Australia in 1999. Tetris DX features battery-saved high scores and three player profiles...
_{Tetris (Game Boy) - Wikipedia}