The bearded drag Queen was due to sing with Basalt, an ensemble of three Syrian artists based in Vienna.
The group were meant to be performing at the opening concert in the New European Song book series.
Conchita, who won Eurovision in 2014, is based in Vienna.
The Austrian singer was supposed to appear in concert at the festival on Friday.
Although the Edinburgh International Festival is a "permit free festival", officials said they have no jurisdiction over the visa process and are not an immigration authority.
A permit free festival means "performers and their legitimate entourages do not need to obtain work permits to appear in the UK".
The Edinburgh International Festival said: "Performers and entourages at permit-free festivals enter the UK as a visitor (standard) and do not need to apply for entry under the Points Based System or as a Permitted Paid Engagements visitor.
"However, you may still need to apply for a visa in advance and the requirements depend on which country you are coming from.
"Please note that legitimate entourage may include directors, producers, technicians, stage managers, creative team, publicists and anyone involved directly with a specific production that has been registered with the Fringe Society."
Eurovision Song Contest winner Conchita Wurst has cancelled an appearance at the Edinburgh International Festival because the musicians she was due to perform with have been denied visas.