Carr died in Los Angeles after complications from a rare form of dementia, her representative said.
In her role in the 1965 film, she famously performed the song Sixteen Going on Seventeen.
After leaving the film industry, Carr ran an interior design firm in California.
Her mother had arranged for her to audition for the role, although she had had no singing or acting lessons.
Rodgers and Hammerstein's The Sound of Music was a massive hit, at the time surpassing Gone with the Wind as the highest-grossing film of all time.
Carr later wrote two books on her experience - Forever Liesl and Letters to Liesl - and frequently appeared at events commemorating the movie.
Her only other major role was in the Stephen Sondheim television musical Evening Primrose.
Many fans of The Sound of Music took to Twitter to post their memories, clipping up sections of her performance.
Kym Karath, who played Gretl in the movie, tweeted: "She has been like a sister throughout my life."
The real life Liesl, Agathe von Trapp, the eldest daughter of the Austrian family who inspired the film, died aged 97 in 2010.

Andronicos Sideras, 55, is accused of mixing horsemeat with beef before selling it on through his company Dinos & Sons.
Two men who ran FlexiFoods - a meat suppliers linked to Dinos & Sons - have admitted passing horsemeat off as beef.
Mr Sideras denies one count of conspiracy to defraud.
The discovery of horsemeat in processed beef products sold by a number of UK supermarket chains in 2013 resulted in a series of product recalls.
The plot to pass horsemeat off as beef only came to light after horse identification chips were found in the meat by inspectors.
Ulrik Nielsen, 58, the owner of FlexiFoods, and his "right-hand man", Alex Beech, 44, have already admitted their involvement in the conspiracy.
Mr Sideras, of Southgate, north London, told the court in that in 2012 he agreed to store horsemeat for FlexiFoods.
He admits he relabelled the shipment because the packaging had been damaged during transit, but denies purchasing, selling or using the horsemeat.
The trial continues.
American actress Charmian Carr, who played the eldest von Trapp daughter Liesl in the film The Sound of Music, has died aged 73.
[NEXT_CONCEPT]
A businessman has denied being part of a conspiracy to pass off horsemeat as beef, claiming he was only storing the product for another company.