The Clarets trailed 3-1 from the away leg of the tie and a 1-1 draw meant a 4-2 aggregate defeat for the Lancashire side.
After enduring relentless pressure from Burnley, Olympiakos took the lead on the night with a goal from Daniel Podence in the 83rd minute. Substitute Matej Vydra scored for the home side three minutes later on his debut for the club.
Burnley's seventh place finish in the Premier League last season earned them an unexpected first taste of European football in 51 years but after wins over Aberdeen and Istanbul Basaksehir they were unable to get past the preliminary rounds.
Despite Burnley needing to overturn a two-goal deficit, Dyche left the club's record signing centre-forward Chris Wood on the bench along with England defender James Tarkowski and influential midfielder Jack Cork.
But the manager said the performance, which saw his team create a series of chances and dominate the Greeks, showed Burnley wanted to progress, despite plenty of talk in the media of the competition being a burden on a small squad.
"I think you saw a team that wanted it to carry on considering how many have been saying that 'no-one wants it', but that is the mentally here, we want to win every game that we play," said Dyche.
"It has been three qualifying rounds but six games. So we have picked up experience from that," he added, saying the challenge of travelling to Europe and then preparing for Premier League football at the weekend was a tough one.
"I think we have learnt a lot. A lot on the pitch, off the pitch, a lot about travelling, about organisation and all the things that come with it. With tonight's performance it could possibly have carried on.
"There have been a lot of learning curves for me, the staff, the players and the club I think. The whole club will learn from this experience," he said.
(Reporting by Simon Evans; Editing by Ken Ferris)
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