When Leonard and Teddy are in the diner, Teddy says "What about John G.? You think he's still here?" When the scene cuts back to Leonard, the line "You think he's still here?" is heard, but Teddy's mouth isn't moving.
While noting down Teddy's license plate number SG13 7IU, Leonard writes it as SG13 71U, which is how it gets tattooed on his leg at Emma's. However, at the beginning of the movie, when Natalie sends Leonard a package containing details about Teddy, the License number mentioned is SG13 7IU. Leonard checks this with the License number tattooed on his leg and does not realize the difference.
In the two scenes where Sammy is administering insulin to his wife, he is placing the needle in the crook of her arm, as if mainlining it. Insulin is never injected at this site, but rather at the back of the bicep. The other shots, in the stomach and the thigh, are correct.
In the Jaguar Teddy gives Leonard the address of the Discount Inn. Teddy met Leonard at this motel earlier and therefore knows that he is already staying there. Leonard did not check out when they left and should therefore have both a picture and the key to room 21 in his possession.
After fleeing Dodd, while driving to the hotel, Leonard is reading the description that Natalie wrote down for him. You hear him 'thinking' what she's written down. He 'says' that Dodd is staying at the Mountcrest Inn, yet it is written as the Monterest Inn.
In the black/white parts Leonard is staying in room 21 (as seen on the key). In one of the shots he looks out the window and we see the door to room 304. In the clip it is at the same height as the room Leonard is looking out of, but when he moves into that room he has to go down the stairs to get to the reception and in doing so passes room 21.
When Natalie shows Leonard the picture of herself with Jimmy, she is smiling (in addition to Jimmy having a mustache). Later, when he gets out of bed and views it again, she is no longer smiling, and Jimmy's mustache is gone. Finally, in the close-up of Leonard writing the note on the back of the Polaroid he took of her, the picture has reverted to its original state.
In the scene where Dodd first comes after Leonard and chases him, his red SUV blocks in Leonard's Jaguar from behind. However, in the next chronological scene (when Leonard gets back to his car and drives away), Dodd's SUV is no longer blocking him in.
Natalie's photograph which has some cancellations and then written something else, the cancellation is very shown as if down by a black marker. However when Leonard is shown making the cancellation on Natalie's photograph it is done using a ball pen and the cancellation is untidy compared to the prior black marker cancellation.
When Dodd pulls his red vehicle into view after Leonard turns off the road to see what he was harassing him about, the camera crew is reflected on the side as the truck moves from the right side of the frame to the left.
The Jaguar XK8 has antilock brakes and would not have managed the entertainingly screeching skid as it stopped outside the tattoo parlor. Only the rear wheels are locked and clearly the handbrake was used to make an amusing noise. No matter how it was achieved, it was not a natural stop and there is no evidence to suggest that Leonard is a stunt driver who might habitually stop like that.
Sammy Jenkins is reported as having "possible damage in the hippocampus, nothing conclusive" but the CT scan of his brain shown at the time is grossly abnormal. It shows a large left sided subdural haematoma (bleed) with significant mass effect. It is very likely that a patient with such a scan would be seriously unwell.
At the end of the flashback where he sustains the injury Leonard seems to fall directly down to the floor after being attacked. His body should then be lying the same way as his wife's. However he lands on the floor with his body facing the opposite way.
Leonard is able to remember that he has a problem with his memory because he has conditioned himself to look at his "Remember Sammy Jankis" tattoo. Similarly he has conditioned himself to check his pockets for Polaroids to remind himself of other essential day-to-day facts. Alternatively, it could be because Leonard has Korsakov's syndrome, sometimes caused by acute trauma. In some cases the patient knows that they have the disorder. Arguments have been presented both for and against this but, in the end, it's the reality that the movie presents us with and so, for 113 minutes at least, it's true.
...or not - your call. Since the movie is about memory and its fragility/unreliability, it is eminently possible that all supposed continuity errors are deliberate. However, given the structural complexity of this movie, and bearing in mind the number of errors which manage to find their way into even the simplest linear narrative, it is equally possible that they are genuine errors and so they are reported here for your consideration.
Natalie's handwriting is different in each of its appearances. The writing on the Ferdy's Bar coaster switches between print and cursive, and the invitation to the City Grill is in another kind of print. However, this could be another demonstration of the instability of Leonard's memory.
Leonard goes to the Discount Inn, and the manager takes him to room 303; when Leonard realizes that it is not his room and the manager admits to also renting him room 304 because business was slow, Leonard calls the manager Burt. In that particular entire sequence, Burt has on no name tag, nor does he mention his name to Leonard.
Leonard has several tattoos on his chest and arms that he would never be able to read clearly himself; they are not reversed so that he could read them while looking at himself in a mirror, nor are they upside down so that he could read them by looking down at his body. The only person who could read them without difficulty would be someone who was staring directly at Leonard, making these tattoos useless to Leonard himself.
Teddy's car's license plate changes throughout the movie. The first time we're able to see it (after Leonard and he leave Dodd's apartment), it reads SG13 71U. Towards the end, after Leonard kills Jimmy, it reads SG13 7IU, which is what Leonard reads it as every time he mentions it. Tattooed on his thigh, it reads SG13 71U, and on the vehicle title Natalie provides for him, it reads SG13 7IU. This, of course, is likely an intentional mistake to reflect Leonard's unstable state of mind.