When Harry leaves Dr Fowler's house after discovering the body, as he retreats down the stairs, wiping his prints, he turns off the hallway light, but the shot shows the light in Dr Fowler's bedroom turning off. It is at the top of the stairs in a room with its own light switch.
When Harry meets Louis Cyphre in the New Orleans church to discuss his progress he states there are 3 dead bodies and he's a murder suspect in two of the cases. Doctor Fowler was killed in New York and there was no mention of a police investigation of his death. And although Margaret Krusmark is dead at this point in the movie, the police had only questioned him about Toots Sweet.
When Harry Angel impersonates an official (Harry Conroy) from the National Institute of Health at the private hospital just outside Poughkeepsie, at the end of his conversation with the nurse, his glasses disappear between the penultimate and final shots of the scene.
When Harry tries to avoid Ethan Krusemark's two goons by buying oysters he ends up on the floor being attacked by their dog. If you look carefully you will see that the oyster sellers hat appears and disappears off his head from one brief shot to another.
After Harry finds Dr. Fowler dead, he lights a cigarette and the camera shows a close up of the gun in Dr. Fowler's hand, not fully grasped, pointing towards the left side of his body. In the next shot Harry goes to grab the Bible with the ammunition inside, and the gun is now pointing up towards Dr. Fowler.
When Harry is running from the goons chasing him, he interrupts a street procession. As he knocks a few people to the ground, you can see that they're wearing knee pads, especially the woman in the skirt. As this woman lifts herself off the ground, you can see her black knee pads.