The Reds travelled to Turf Moor knowing at least a point would take them to the top of the Premier League, and got off to an electric start through Darwin Nunez's brilliant goal on six minutes.
His goal was assisted by Gakpo and the Dutchman thought he had scored for himself later in the first-half.
However, his finish was not allowed to stand as referee Paul Tierney spotted a foul in the build-up.
Nunez was adjudged to have fouled Charlie Taylor inside the area and VAR also took a look, with the goal remaining chalked off.
However, Liverpool fans on social media were angered and one took to X to say: "What's wrong with the goal?!? Gakpo robbed."
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"Very soft there, Gakpo goal disallowed," added another.
A third, with laughing emojis, said: "Gakpo should have a goal and two assists in 33 minutes."
"Why was Gakpo goal disallowed because VAR showed that Nunez didn't touch the Burnley player," said a fourth.
The Reds went in at half time 1-0 up having missed a catalogue of chances to double their advantage, including Mohamed Salah's strike rattling Clarets goalkeeper James Trafford's crossbar.
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VAR intervened once again at the start of the second-half as Harvey Elliott's goal was ruled out for a Mohamed Salah offside.
However, replays appear to show he was pushed into an offside position and once again, Liverpool fans expressed their frustrations.
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"Salah is clearly being pushed into that offside position. I feel sick," said one.
A second added: "Goalie is never ever ever saving that in a million years and Salah is pushed into an offside position by the Burnley player anyway. Absolute f****** joke."
"There's no way you can rule that out! F*** off. Has nothing to do with it and he's being pushed there by the defender," said a third.
A fourth asked: "Salah was pushed offside… why didn't they show that on the VAR monitor."
At the third time of asking, Liverpool scored their second goal in the 90th minute when Diogo Jota netted from a tight angle.