Armadillo tells Rango to follow his shadow, which means he should be going opposite the direction of the sun, yet you see as Rango walks away that his shadow is pointing out to his right, indicating that the sun is to his left.
While Rango hangs out in the Mayor's antechamber, he plays with a golf club, and knocks over a couple of actual golf balls that fit his character size. Yet when he plays golf with the mayor later, they both use pill bugs as golf balls.
At the accident, when the fish tank starts floating, Rango has the glass in his left hand, but when he gets up and stares at the toy fish, it is in the right. In that same scene the doll's arm goes from left to right as well.
When Rango stares down Jake ready for a gunfight, a dragonfly flies by and Rango catches it with his tongue and eats it. Dragonflies are located around abundant water resources and the town of Dirt is bone dry.
The clock tower is prominently shown to be a large (by their scale) ticking Telechron pocket watch, complete with company logo on the dial. However, Telechron never made either pocket watches or wind-up mechanical movements; their entire product line over 80 years consisted of electric clocks and related electrical timing devices, all powered by motors.
When the mayor pulls the trigger to shoot Rattlesnake Jake, we here a ratchet sound, and the gun's cylinder rotates to....an empty chamber. The gun is a single action Colt Peacemaker (the classic western gun), and the cylinder would not move on the trigger pull, it would move to position when the hammer is cocked before each shot.