Megan Lee, from Oswaldtwistle, Lancashire, died on 1 January, two days after she was admitted to hospital, Lancashire Police said.
The force said it was carrying out inquiries on behalf of the county's coroner.
The takeaway involved, in the Hyndburn area, has not been identified.
A post-mortem examination has been carried out but police said the full results would not be known "for some time".
Lancashire County Council said its trading standards department was assisting with the investigation.
Megan's parents Adam and Gemma Lee described their daughter as "a role model, an inspiration, our princess".
In a statement, they said: "Our hearts are shattered at the sudden loss of our beautiful daughter.
"Megan loved school and thoroughly enjoyed drama and musical theatre.
"She was the kindest, most loving daughter and sister who always wore the most beautiful and infectious smile."
Richard Jones, head teacher at Megan's school, St Christopher's CE High School in Accrington said: "Megan was a super student, a wonderful pupil at our church school and we miss her terribly.
"It feels as if a member of our family has left us. She was bright, articulate, somebody of real potential who had thrived and grown during her five years with us."
A 15-year-old girl has died after suffering a severe allergic reaction to takeaway food, prompting a police investigation.