Homing pigeons can only fly in one direction, not round trips.
When Det. Tripp learns about Finch's true identity and looks at his fake passport, one of the dates on it is 29 FEB 95. There was no 29th of February in 1995.
Witnesses in protective custody are not allowed in front of windows for fear of snipers.
When Tess is on the roof of the building with the gun, the magazine she inserts is shown to be empty when she pulls back the charging handle.
When Finch and Micah are making their new identification papers, there is a shot where the corner of laptop that Finch is using to make the ID's is visible. The green screen on the laptop screen is visible - no post-production was done to the computer screen in that shot.
During the father-daughter train ride to hide the box (set in the 1970s), the two locomotives shown are in CSX paint. The railroad CSX was not formed until July 1, 1986 as a renaming of the Seaboard System Railroad and Chessie System into one entity. CSX never had passenger service.
Tess is not wearing gloves when she fires the rifle, and she abandons it on the rooftop in plain sight. Since she was, by her own admission, a child raised by the Foster Care system, her fingerprints would be on record. A police investigation would arrest her in no time at all.