Members of the Hibs Supporters Association had appealed to Dempster for permission to hold a rally inside the stadium's East Stand on Monday.
They are attempting to heap more pressure on unpopular chairman Rod Petrie following their humiliating relegation from the Scottish Premiership.
But their request was turned down by former Motherwell chief Dempster on Tuesday, who only officially started work at the Leith club the day before.
Former Hibs midfielder Paul Kane - who is leading the 'Petrie Out' campaign - said: "We are very disappointed the club is shutting the doors on its own fans.
"For health and safety reasons we asked for access to the East Stand, which holds 6,000 people, rather than have a large crowd gather outside the ground.
"Leeann Dempster has cited renovation work along with logistical reasons and the timeframe as reasons for turning us down.
"We are sure these could easily have been overcome and we could have worked together to sort it out.
"The health and safety of Hibs fans on the day is paramount. We will speak to the police and the council about it.
"We did contact Leeann Dempster at the earliest opportunity, as soon as she took up her post at Easter Road yesterday morning.
"We are also disappointed that we heard our request had been turned down through the media - not from the club.
"It would have shown some courtesy and respect to the fans to have contacted us first.
"It is another PR own goal but you get used to that. All too often the supporters seem to be an afterthought."