No. I was in Rome and Vatican City in Sept. 2022: Nobody asked for a passport to cross the "border".
If you want an interesting border-crossing experience, think about visiting the Archbasilica of Saint John Lateran, which is relatively close to the Colosseum. It's the first "Vatican" (consecrated in 324) but lies outside of Vatican City. In front of the church, there are a series of bollards to mark the transition from Italy to Vatican. Visitors cross borders when they enter St. John Lateran just like they do when visiting Vatican City. It's the only archbasilica in the world.
Beyond the border-crossing trivia, it's a stunning church that was built over a period of 1,400 years. If you visit, make sure to see the Holy Stairs in the baptistry, which stands apart from the archbasilica. The stairs that form the staircase once led to the praetorium of Pontius Pilate in Jerusalem and which, therefore, were sanctified by the footsteps of Jesus Christ during His Passion.
More broadly, I visited five city churches in Rome on my most recent visit. There are some incredibly beautiful ones: You don't where to look because they are so stunning. See Rick's book for his church tour, which lists five or so churches outside the Vatican worth visiting.