The Quakers, currently fifth in the National League North, do not have the 500 covered seats needed across two stands at their Blackwell Meadows home.
Clubs were previously able to install temporary seating with conditions, but the rules changed in May 2016.
Now, permanent seating must be in place for clubs to meet the criteria.
The club said it would liaise with National League South side Poole Town, also fifth in their league, who have similar issues with their ground.
"This change in the requirement for permanent covered seating was only brought to the board's attention in the past couple of months," a statement from Darlington said.
"By the time the board was fully aware of the requirement, it was not able to act because of the timescales involved and the lack of funds available.
"In order to be able to meet the seated stand requirements, an additional round of fundraising through the Community Share route has a lead time of at least six weeks to prepare the offer and then an unknown amount of time to raise the estimated requirement of £150k, plus the actual construction time for the additional seats.
"We understand that an absolute minimum of 12 weeks would be needed - time that just wasn't available once the club had finally moved into Blackwell Meadows."
The decision comes as a blow to Darlington, particularly given the optimism which followed their return to the town in December after a spell at Bishop Auckland's Heritage Park - the first time since the old Darlington club had been liquidated in 2012.
Should the appeal not be successful, the next ambition is to put plans together for redevelopment work that does make the stadium sufficient.
"Understandably, our manager, Martin Gray, is very frustrated at this situation, as the incredible efforts of the players and coaching staff will not give us the chance of promotion this year unless the appeal is successful and we are allowed to participate in the play offs," the statement continued.
"We are as disappointed as everyone else that our success on the field is not reflected in our ability to move to a higher level at this time. But we would remind everyone that had we thought that we would be playing back in our home town, in the second tier of the non-league pyramid, under our original name, and all within five years of that fateful day in 2012, not many would have had faith in that dream.
"The board is now working with the Supporters Group to set a five-year-plan to develop the ground at Blackwell Meadows, in conjunction with Darlington RFC, which we hope will meet the ambitions of the manager and players and playing staff, all of our volunteers, the board, and most importantly of our fans and the people of Darlington and the surrounding area."

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24 September 2015 Last updated at 07:04 BST
He is now the record holder in the long and high jump, the shot-put, discus and 100 metres sprint.
Pellman ran his 100 metres in 26.99 seconds.
He also became the first person over the age of 100 to clear a bar in the high jump, eventually reaching 0.90 metres.
Pictures courtesy of Ken Stone/TimesOfSanDiego.com
Darlington will appeal the National League's decision to deny the club a play-off place because their ground does not meet seating requirements.
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100-year-old Don Pellman has set five world records in his age group at the San Diego Senior Olympics in America.