Paintings and Victorian marble statues held at Carbisdale Castle, a former youth hostel near Ardgay, have been valued at a total of ??500,000.
The art is to be sold at an auction in London next month.
Organisers of the petition on the website change.org want the works should remain in the local community.
They said because the castle and its contents were gifted to SYHA, the association should not be selling off the art to make a profit.
More than 700 people have signed the petition so far.
SYHA shut Carbisdale Castle in Sutherland four years ago because of rising repair costs.
A former stately home, it was known as Castle of Spite because of the part it played in a family feud. The property is also said to be haunted by a ghost called Betty.
SYHA said it would invest money raised from the sale of the castle's artworks in its network of youth hostels.
The 17 sculptures and 36 Italian and Scottish 19th Century paintings will be offered for sale at an auction in London on 20 May.
Auctioneers Sotheby's have put a combined value of ??500,000 on the art.
White marble sculptures from the 1800s to be sold include Andromeda by Florentine sculptor Pasquale Romanelli and Nymph at the Stream by Scottish artist David Watson Stevenson.
It will be the first time in more than a century that the Victorian sculptures have been put on the market.

He was travelling on a United Airlines flight from Rome to Chicago on Saturday that was diverted to Belfast International Airport.
The 42-year-old, who has dual US-Italian nationality, has also been charged with disruptive behaviour on board an aircraft and common assault.
He is due to appear before Coleraine Magistrates Court on Monday.
The plane resumed its journey on Sunday afternoon.

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The midfielder agreed personal terms with the Canaries this week and, having passed his medical, he has penned a three-and-a-half-year deal.
"He can add goals to our midfield, which I think we need," said Norwich manager Paul Lambert.
"He'll get better as a player the higher the level he goes up. If he does that he'll be a great player for us."
Former England Under-21 international Howson, a product of the Leeds academy, has made almost 200 appearances since his debut in 2006, but his contract was due to expire in the summer.
He has featured 22 times for his home town club this season, although he has not played since picking up a knee injury in the 2-0 win over Millwall at the beginning of December.
Meanwhile, Lambert has outlined a back injury as the reason for terminating the season-long loan of Manchester United defender Ritchie de Laet five months early.
The 23-year-old Belgian has played seven times for the Canaries, but has not featured since the 2-0 Premier League loss to Tottenham on 27 December.
"Ritchie's done great for us," said Lambert. "But he has a back injury which was keeping him out for periods of time."

"I was expecting it to be quiet, but not that quiet. Even the homeless people were still asleep," Dean Saunderson said of his expedition to capture a unique side of a city.
The 21-year-old photography student had been looking for inspiration for his project to make a portrait of his home town, Nottingham.
"I came across lots of images of abandoned towns and cities, completely untouched by humans for years, " he said.
"Knowing there isn't anywhere like that in the UK, I wondered if I could get close in some way."
Getting up before dawn on Christmas Day, he found a city given over to the birds and litter.
"It was a big shock. It felt so different, eerie. I kept checking over my shoulder.
"There was no one looking at you but it felt like there was no one looking out for you either - like anything could happen to you and no one would know.
"It reminded me of those pictures of the city near Chernobyl."
His ambition is now to visit a truly deserted place and put together a photo essay chronicling its story.
A petition has been started opposing the sale of artworks by the Scottish Youth Hostel Association (SYHA).
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