In the history of Formula 1, when has the fastest qualifier not started the race on pole?

This has happened twice more since Ste's answer:

• 2021 Monaco Grand Prix

Charles Leclerc took pole position, but crashed at the end of the session. The crash caused damage to his driveshaft that wasn't discovered until he left the garage on Sunday, leaving no time for it to be fixed or replaced, and he did not take part in the race.

• 2021 Italian Grand Prix

Valtteri Bottas was fastest in qualifying and won the sprint qualifying session, but engine penalties meant he started the actual race at the back of the grid, and Max Verstappen took pole instead.

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Taking the title of the question literally, you could also count the 1997 European Grand Prix. Jacques Villeneuve took provisional pole with a time of 1:21.072, but Michael Schumacher and Heinz-Harald Frentzen both matched Villeneuve's time down to the millisecond. Villeneuve was awarded pole position by virtue of having set his time first, so Schumacher and Frentzen missed out on pole despite being (joint) fastest qualifiers.

I was so interested in this myself that I wasn't going to stop at anything to get a good answer. Sadly, small research didn't get me the answers I was after so eventually I spent 14 hours reviewing the reports of every Grand Prix since 1950. To the date of this answer there have been 864 of them, which is why it took so long.

Grands Prix in which the fastest qualifer didn't start from pole

1955 Indianapolis GP - May 30, 1955

Jack McGrath set the fastest qualifying speed but actually started in grid position 3, on the outside of the track. Jerry Hoyt started in pole as he was one of only two drivers to complete a qualifying event on "Pole Day" due to strong winds and the threat of heavy rain. Hoyt was actually only the eighth fastest in the field and his speed (140.045mph) is the record slowest ranked pole speed.

1994 Monaco GP - May 15, 1994

This was the first GP to follow the deaths of Ayrton Senna and Roland Ratzenberger at the season's previous Grand Prix at San Marino.

Whilst Michael Schumacher did start at pole position and is officially documented as such, grid places 1 and 2 were left unoccupied and painted with the flags of Senna and Ratzenberger's respective nations as a mark of respect.

1996 French GP - June 30, 1996

Michael Schumacher qualified with the fastest lap, earning him pole position for this race. However, Schumacher's engine failed during the formation lap at Magny Cours and did not take his place on the grid.

1998 Japanese GP - November 1, 1998

Schumacher was involved again here, this time stalling from his hard-earned (one tenth of a second faster) pole position. As a result of this he was sent to the back of the grid. The race started successfully on this, the third, attempt after Jarno Trulli had stalled his car on the first attempt.

2001 Monaco GP - May 27, 2001

David Coulthard was sitting happily in pole position but his McLaren failed to pull away for the formation lap. His team managed to get him moving eventually but by this time he was resigned to starting from the back of the grid.

2005 US GP - June 19, 2005

This is the one they called "IndyGate". Jarno Trulli was fastest qualifier and therefore in pole position. However, due to safety concerns regarding the Michelin tyres, all drivers of vehicles with these tyres retired to the pits during the formation lap.

Only six cars started the race with drivers in grid places 5, 7, 17, 19, 18 & 20.

2005 Italian GP - September 4, 2005

Kimi Räikkönen qualified with the fastest time but was given a 10-place grid penalty for changing his engine. Juan Pablo Montoya took up pole position as a result.

2007 Hungarian GP - September 4, 2005

Fernando Alonso recorded the fastest time in qualifying but a stewards' inquiry before the race adjudged Alonso to have impeded Lewis Hamilton during Q3. Alonso was subsequently relegated 5 places on the starting grid, moving Hamilton up to pole.

2012 Spanish GP - May 13, 2012

In this event, Lewis Hamilton was excluded from the qualifying results due to having insufficient fuel to return to the pits. As a result, Hamilton had to start the race from the very back of the grid.

2012 Monaco GP - May 27, 2012

The fastest qualifier at Monaco was Michael Schumacher. However, Schumacher was carrying a 5-grid penalty from the previous race in Spain for causing an avoidable collision with Bruno Senna, thus moving him back in the grid.

It nearly happened in

1990 Japanese GP

Ayrton Senna had secured pole position but was unhappy that pole was situated on the side of the track which was not on the racing line. Senna requested that he be moved and the stewards agreed, only for the FISA president to overrule the decision, meaning Senna started in pole at the original location.

And to close, you ask:

There is a reason for the question as the Fantasy F1 game I run gives the pole-sitter bonus points but I am not sure what to do in this kind of situation!

You'll have to decide! Perhaps give the bonus for fastest qualifier instead of pole position?

You can see my spreadsheet with all the pole positions since 1950 here.