The Ligue 1 champions - who had Kylian Mbappe sent off late in the game - were beaten 9-8 on penalties following a 2-2 draw at Stade de France as their hopes of a domestic double were dashed.
Neymar was then videoed attacking a fan on his way to collect his runners-up medal as the PSG team were ridiculed by the jubilant Rennes support.
The French Football Federation have now announced that the world's most expensive player will be forced to miss three domestic matches, with a further two matches suspended.
While having to plan without Neymar at present, PSG have stated their intention to appeal the sanctions handed out to the South American.
A statement released by the club on their official website read: "Paris Saint-Germain takes note of the decision of the Disciplinary Committee of the French Football Federation (FFF) to sanction Neymar Jr with a suspension of 5 matches, of which three are set.
"Given the insults suffered by several players of the Club including Neymar at the end of the Coupe de France Final, and the various elements transmitted by Paris Saint-Germain to the Commission, Paris Saint-Germain considers this sanction hard.
"Paris Saint-Germain and Neymar Jr have decided to appeal this decision to the Higher Appeal Commission of the FFF."
If the suspension is upheld, then it will bring to an end a disappointing campaign for the Brazil international, who had hoped to kick-on after injury saw him miss the second half of his first season in the French capital.
But despite an encouraging first few months under Tuchel's stewardship, a foot injury picked up against Strasbourg in January saw his season grind to a halt at a far from ideal time.
Despite holding a 2-0 lead from the first leg at Old Trafford, the Parc des Princes outfit were beaten 3-1 in the return fixture after a controversial penalty - awarded for handball by Presnel Kimpembe - was converted by Marcus Rashford in injury-time.
An angry Neymar posted an expletive-laden message on Instagram in protest at the performance of the officials; an action that recently saw him pick up a separate three-match European suspension that will see him miss PSG's opening group games in next season's Champions League.
In their star turn's absence, PSG did go on to seal the Ligue 1 title, though they stumbled over the line somewhat with Tuchel's side currently on a run of one win in seven matches across all competitions.
That sequence included a 5-1 humbling at second-placed Lille as well as the cup final defeat to Rennes which meant they failed to win either of the domestic cup competitions in France for the first time since the 2012-13 season.
Neymar will now turn his attention towards the summer and winning the Copa America on home soil with Brazil, with the Selecao getting their tournament under way against Bolivia on June 14.