The energy supplier will not extend its current four-year contract, believed to be worth about £32m, with the Football Association after the final next May.
The company says the decision not to renew is because it intends to pursue a new sponsorship strategy.
The FA says it will now offer the “FA Cup sponsorship opportunity to the open market” with a four-year deal from next year.
As part of the sponsorship, Eon had tried to reduce the carbon footprint of supporters during each Cup campaign by encouraging fans to cut their own energy emissions.
It had hoped that the energy saved would go some way to offset the 45,000 tonnes of carbon dioxide produced during the FA Cup competition each season.
The FA is still looking to agree a broadcast deal for the games that were to be shown by Setanta following the collapse of the broadcaster earlier this year.