Stanley, whose worldwide fame far outweighs anything they have achieved on the pitch in their chequered 118-year history, face becoming the first club to go out of existence twice during a league season after the demise of the original team in 1962.
The Lancashire club was reformed in 1968 and regained its League place in 2006.
Now it has been ordered to settle a tax bill of 308,000 pounds owed to the Inland Revenue in the next seven weeks, or else be forced out of business.
Premier League Burnley, who are back in the top flight this season for the first time since 1976, played Accrington at Turf Moor on Tuesday night when 5,000 people raised about 50,000 pounds for the club.
Stanley are widely known internationally because of their unusual name, their 1962 collapse and a name check in a famous 1980s TV advertisement for milk - which has been downloaded more than half a million times on the Internet.
The club was formed in 1891 as Stanley Villa, named after the Stanley Arms pub on Stanley Street where the players met, and adopted the Accrington town name in 1893.
Burnley manager Owen Coyle said: "Burnley have known hard times in the past and we will do whatever we can to keep Accrington alive."
Stanley gave a "huge thank you to Burnley" on their website, adding Tuesday's match, which Burnley won 4-0, was "a magnificent display of footballing solidarity."
