Colour Sergeant 'Tats' Taylor, shared pictures on Facebook posing next to a Red Arrows jet with Corporal Shane Woods and Private Anthony Panton.
The 49-year-old posted the images with the caption 'Think we took over RAF Scampton' and a laughing emoji.
Senior figures are said to be furious with the pictures which were taken on Friday, at the historic home of The Dambusters after a boozy party, The Sun reports.
The aerial display plane is parked next a plaque marking the deaths of Flight Lieutenants Jon Egging and Sean Cunningham, who died in separate accidents in 2011.
Sgt Taylor's pictures showed all three men, all of 2nd Battalion The Royal Anglian Regiment, n.a.k.e.d. except for regimental headdress.
A source told The Sun: 'Tats put the pictures up on Facebook but took them down after, but it was too late, everyone had seen them. They know they’re in the s***.'
It is claimed the three soldiers had been given alcohol as a token of appreciation by the local council after spending 10 days at the mobile Covid-19 testing site at the Lincolnshire base.
Friends of Sgt Taylor told The Sun he did not realise the significance of the plane.
An Army spokesman said: 'This behaviour clearly contradicts the discipline we ask our personnel to display and it undermines the support we provide to health authorities with mobile testing and other tasks across the UK.
'We are looking into this incident and will deal with the matter accordingly.'
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