When the mob is preparing to send back Old Joe, there are four gangsters. Two restraining him, one attaching the gold bars and another preparing the time machine. In the fight that follows, there are only three of them.
When Joe is talking to Cid in the kitchen at night, Cid has a noticeable gap in one of his front teeth. In the next scene Cid is talking to his mother Sarah and still has a very wobbly baby tooth where the gap was previously.
When Joe and Cid are having a conversation looking out of the hatch at the end of the tunnel from the house, the branch holding up the hatch changes position from shot to shot. It is rotated about 90 degrees between the positions and the position changes four or five times during the conversation.
When Joe and Old Joe are in the café together, the waitress Beatrix comes over to take their order, and their table is empty. However when she returns to give them their food, there are two drinks already on the table. We know that she has not given them their drinks order previously because the rest of the scene is a conversation between the 2 characters in which she does not reappear.
Before Joe's first meeting with Abe, Kid Blue is messing around with his gun, eventually winding up pointing it at Joe and cocking the hammer. The film cuts to a shot of Joe, then to a shot of Kid Blue pointing the gun at Joe with the hammer uncocked. The next shot of Kid Blue shows him with his thumb on the cocked hammer.
When Cid is answering multiplication questions with Sara and answers the 7x3 question for the 21 tile, which is then placed on the board, it disappears in the next shot, reappears in the following shot and disappears again in the next shot.
When the truck is upended and overturns, it is clear that there is no one in the cab. Then when it comes to rest upside down on the road, the close-up reveals the Sara and Cid characters held inside by their seat belts.
When Sarah is doing multiplication tables with Cid, the first tile we see him place (56) moves repeatedly between shots - it's correctly on 7x8 from his point of view, but the tile is on 7x9 when we move to her point of view. The next tile (32), which he supposedly places on 8x3, is actually placed on 7x8, adjacent to the incorrectly-placed first tile from Sarah's point of view.
When Joe and Seth are talking in Joe's apartment towards the beginning of the movie, the streams of tears on Seth's cheeks and upper lip change multiple times in between the close-up shots of his face.
Joseph Gordon-Levitt went through subtle prosthetic makeup around his nose, lips, and brow to look more like Bruce Willis, but the makeup artists failed to synchronize the actors' ears. Willis has lobed ears while Gordon-Levitt has attached ears.
Young Joe disposes of bodies, by throwing his victims downwards a shaft, ending up into the fire. The actual body we see ending up into that fire seems to slide horizontally towards the fire, instead of vertically falling into it, given the orientation of the fire.
When the younger version of Joe's older self is waiting for HIS older self to appear, stubble is shown clearly on his face. After he shoots his older self he bends down to collect his silver/gold bars and the stubble on his face is gone.
When Seth fails to close his loop and is on the run, Abe tells the gat men it would be preferable not to kill Seth because all sorts of bad (time travel paradox related) things could happen if they did that. However, by cutting off all his limbs in the present is in effect as good as killing him because as they keep him on life support the next thirty years, Seth will be unable to do the things he would otherwise have done, which changes the future in the same way as if he had been killed. Also, he would not have been able to run anywhere once he would have been sent back from the future with no limbs, starting another round of grandfather paradox.
There is a "grandfather paradox" inherent to many stories about time travel. When Young Joe "closes his loop" by killing himself, Old Joe never existed, so he can't be sent back in time. Young Joe does this so that Cid never sees Sara killed, so Cid doesn't become the Rainmaker. This would undo all events directly involving Old Joe in the movie's present: the damage to Sara's farm, Abe and the Gat Men being killed (they'd all be alive again), Cid's cheek would never have been injured by Old Joe's bullet, and the gold bars wouldn't be lying on Sara's farm, etc. In fact, Young Joe would never have met Old Joe at the restaurant and never would have gotten the map that led him to Sara's farm, so he'd never have met Sara and Cid. Young Joe would never have learned that Cid would become the Rainmaker, nor that Joe's older self would be responsible. This means Young Joe would never have killed himself in the first place; he'd move to China for retirement, grow into Old Joe, and the "loop" would begin all over again. In Looper, this is not necessarily a plot hole. The first time we see Young Joe fall from the fire escape, the film cuts straight back to the field where is he is supposed to kill his older self to close the loop, and he does so. At that point of the film the implication appears to be that Joe dies in the fall, so his future is changed and Old Joe never escapes and the events we've just seen never happens. It turns out that he didn't die in the fall, but he does die later. The whole thing is intended to be circular - a loop.
After Cid blows up Jesse at the farm and runs into the field, as Joe approaches him, his face is full of blood. Yet, in the blood, you can clearly see the outline of goggles or glasses that were used to protect his eyes.