The singer started a diploma at the Academy of Contemporary Music in Guildford, Surrey, in 2009 but gave it up as his musical career took off.
A letter formalising his resulting failure was sent to him a year later.
It is on show at Christchurch Mansion, Ipswich, along with other personal objects and memorabilia.
Sheeran's father, John, has curated the Ed Sheeran Made In Suffolk exhibition, which opens on Tuesday.
In an accompanying book, Mr Sheeran said his son had become "disillusioned" with the course after just three weeks. Sheeran was then asked to be a support act for Just Jack, best known for his single Starz in their Eyes.
"He then rang us [along with his wife, Imogen] to say that he was determined to go, even if it meant him leaving college," Mr Sheeran said. "Neither of us were surprised. Ed was developing a fearless gut instinct."
Posters from the tour with Just Jack are hung alongside promotions for other gigs around Suffolk and Norfolk, where Sheeran made his name.
There are also photos of his time at Thomas Mills High School in Framlingham, Suffolk, where Sheeran was was voted "most likely to be famous".