Elche, who ended the campaign in 13th place, have been punished for "very serious infractions" of Spain's sports law and league rules and fined just over 180,000 euros ($202,700), the LFP said on their website.
Based near Alicante in south east Spain, Elche have already said they would appeal if they were relegated over their tax debts, which they have insisted are due to be settled in full by the end of July.
The club owes around 4 million euros, local media have reported.
"This is the biggest injustice that Spanish football has committed," Elche president Juan Anguix told reporters. "We will take the fight to be proved right as far as we can."
Eibar were playing in La Liga for the first time in 2014-15 and went down on the final day of the season after finishing 18th. An LFP spokeswoman confirmed the Basque club would replace Elche next term.
($1 = 0.8880 euros)
(Reporting by Iain Rogers, editing by Sudipto Ganguly and Justin Palmer)
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