Van Persie's 30 Barclays Premier League goals helped propel the Gunners back up into third place and secure Champions League football at the end of a testing campaign at the Emirates Stadium.
The 28-year-old Dutchman has not yet signed a contract extension to a deal which runs out in 12 months' time, with talks between Van Persie, manager Arsene Wenger and chief executive Ivan Gazidis said to have been inconclusive as the player now prepares for Euro 2012.
Former skipper Cesc Fabregas was sold to Barcelona last summer, while contract rebel Samir Nasri eventually also left in a big-money move to Manchester City.
Vieira believes Arsenal and Van Persie are at a crossroads.
"Losing Samir Nasri and Cesc Fabregas last year was a big shock for Arsenal. If they lose Robin Van Persie this year it will be a drama," said Vieira, who was in London to help launch ITV Sport's coverage of Euro 2012, where he will be a pundit.
"With Samir and Cesc you can say 'okay those two are gone', but behind you know that there is Jack Wilshere and Van Persie still at the club - but if Robin comes to leave, that means next year it may be Theo Walcott, then the next year is it going to be Wilshere? It is going to be difficult for Arsenal to keep the top players," he added.
"On the other side if they keep Robin they are going to send a positive message to all the clubs and to all the people around and I think it's really important to keep him."
Vieira, who won three Premier League titles and the FA Cup four times while at Arsenal and is now football development executive at Manchester City, continued: "From Arsenal's point of view, I think that is something they should have closed a long time ago, but Arsenal don't need me to tell them that.
"That is the way it is and they will have to deal with it.
"It is a big risk because they lost Samir and Cesc last year. You never know and will have to wait and see what happens."
Gunners boss Wenger has already moved quickly to bolster the squad for next season, bringing in German striker Lukas Podolski from Cologne.
Vieira feels further impressive signings could help sway Van Persie.
"Robin wants to win trophies. Arsenal have to show him he can do it with Arsenal," Vieira said.
"The best way for them to get him to sign a contract is to bring players to the club who can show him that the club are in the same line as him.
"It is really important because we know how Robin loves Arsenal - that is not the question, the question is how Arsene Wenger will convince Robin they can win trophies."
City have been touted as one potential destination for Van Persie should he look to leave, with former team-mates Gael Clichy and Kolo Toure also already at the Etihad Stadium.
Vieira believes City's first championship in almost half a century is only the start for the club backed by the billions of Sheikh Mansour.
"It is a new generation who want to write history at the football club," he said.
"They don't want to stop here, they want more success and I think that is what makes it really exciting for the next few years.
"I am not just talking about the players and the team, I am talking about the football club, the brand.
"The football club is in a different level now and we haven't achieved what we can achieve yet."