Rory, Paris and Francie return to Chilton as guests for an "alumni event". Tristan is also there, but he shouldn't have been, as he's not a Chilton alumnus - his father had sent him away to military school in North Carolina in Season 2.
When Rory and Paris are at Chilton, Paris runs into a bathroom after seeing Tristan. The bathroom she runs into, and subsequently kicks some girls out of, has a round door knob. Also, looking in the background when the girls are kicked out, it's pretty obvious that the room is dark. When Rory follows Paris into the bathroom there is no longer a knob on the door and the room is all white and very bright.
When Rory calls Logan from New York, the clock on her phone says 5.05, yet she apologizes for calling at 2 am. The time difference between London and New York is only five hours, so the time in London should be 10.05.
In addition to incorrectly listing multiple territories as independent countries, Kirk and Taylor also respectively mispronounce "Kiribati" and "Qatar".
When Rory is giving her speech to the classroom at Chilton, the clock on the wall shows a seconds, minutes and hour hand, but the seconds hand isn't moving, so the time is stuck at 7:50:09.
At the food festival Taylor mentions that all 195 countries should be represented. That number is correct in terms of independent countries of the world (UN member and observer states). However, afterwards it turns out that also dependencies and territories like Guam, Turks and Caicos Islands, Tibet and the U.S. Virgin Islands should have showed up, in which case the number of planned participations would have been higher than 195.
When Paris and Rory meet in the Chilton courtyard, Rory is wearing a wedding band and engagement ring on her right hand. By the time they meet Headmaster Charleston in the lobby, they have gone. In real life, Alexis Bledel is married, so she has clearly forgotten to take them off for the courtyard scene.
As Rory and Paris exit the classrooms at the same time, Paris comments on the "synchronicity". She actually means synchronisation. Synchronicity is an entirely different concept altogether, and surely something someone as allegedly highly educated and particular as Paris would know.