At the beginning of the movie, Nick's red car collides with a locked metal gate and breaks it open. In the next shot, the car is undamaged.
2 minutes, 40 seconds into the movie Ryan Renoylds is planting what it looks like a small Apple tree. It's red and shiny like apples. Next scene shows them talking about how he planted an orange tree and cut to a small orange tree planted through the window. You can definitely tell the difference through the whole scene.
When Nick goes to find Julia on her evening jog at the high school, we first see her jogging in lane 3 of the track, just rounding the corner. The following second, we see her from another angle showing her in lane 4 and about 60 yards farther down the track.
It's established that the RIPD see themselves in mirrors as everyone else sees them, not as the people they were when they were alive, however when Roy is trying on hats at the baseball stadium, he sees himself in the mirror as Roy - as he was when he was alive.
When Roy and Nick race a yellow cab, the car starts with no damage, then has slight damage to the driver side front wing, and then when the chase starts has heavy damage to the front bumper grill and hood. All happen without any collisions occurring.
In the elevator scene, Nick shoots the elevator cable that causes the elevator to rocket upwards. If the safety features are defeated, shooting the counterweight would send it downward. Sending it upward by shooting a counterweight would imply the weights are there to keep the elevator from rising. Also, when the elevator breaks through the roof, there is no sign of the rigging and machinery the elevator needs above it.
At the actual weight of gold, the Staff of Jericho would have collapsed the roof of the building. However, it is plausible that the Staff pieces are not solid gold, but have something else inside (something much lighter), which may explain how they can function as they do.
When Bobby freezes the R.I.P.D. station, he brags about how the device freezes them but does nothing to "us". Why? Everyone involved is dead but hanging around afterwards. It can't be because they've been "popped" because Bobby hasn't popped yet but he's included in not being frozen.