No, Lassie did not die in the movie.
The original Lassie was not a female as the films would have us believe, but rather a male Rough Collie, named Pal. In fact, Pal was initially rejected for the title role in the original 1943 Lassie film, "Lassie Come Home," due to his gender. He was taken on as a stunt dog instead, while a female Collie was hired for the main part. Pal, however, reportedly impressed the filmmakers so much during production that they ended up releasing the female Collie from her contract and using him instead.
After "Lassie Come Home," Pal starred in six other MGM Lassie films from the mid-1940s to the early 1950s, as well as the two pilots for the "Lassie" television series filmed in 1954. For the original film, Pal earned a salary more than twice as large as his co-star, Elizabeth Taylor.
Pal died in June 1958 at the age of 18, after suffering from deafness and blindness during the last year of his life. Pal's trainer, Rudd Weatherwax, reportedly slipped in and out of depression for years after his beloved dog's passing.
See The Most Famous Dogs in Movies and Television at Time.com.