Many of the continuity errors listed here may well be deliberate. In the DVD extras, the movie makers explain that they played with the continuity to underline the struggle between the doctor and technicians trying to erase the memories and Joel trying to hold on to them.
When Mary first comes to Joel's apartment to visit Stan and Patrick, she goes into the kitchen to pour drinks. Just after she says "Let me do the honors," the pendulum of the clock on the wall is not moving. About 2 seconds later, as she is pouring the drinks, the pendulum on the clock is swinging.
In the scene where Mary first comes into Joel's apartment, Patrick and Stan had been drinking beer. At first, just before the door is opened, they are both Rolling Rock (green bottles). After she has entered, Stan's beer turns into a Labatt Blue (brown bottle).
When Clementine and Patrick are driving back from Boston and Clementine is clearly upset, Patrick tries to comfort her by praising her with compliments. Once he calls her 'nice', she gets even more upset, which seems to call back to her "memory" of her and Joel discussing 'nice' on the train. However, that happens the next day, after the erasing happens, when Joel and Clementine both go to Montauk. Unless the word 'nice' had had a similar meaning in their previous relationship, as they are potentially meant to be repeating the same relationship over and over.
In the beginning of the movie, as Clementine is walking on the beach, you can see her take a couple of steps on the wet sand near the water, but in the next shot, as Joel is looking at her, she is standing on dry sand.
While on the train on the way from Montauk point, Joel and Clementine are shown to ride the train to their destination of Rockville Centre, NY but the Montauk branch of the Long Island Railroad only spans as far as Babylon, NY (a full 12 stops away from Rockville Centre, NY).
Joel says that he is going to "Rockville Center", spelled that way in the DVD English subtitles. There is a real village with that name on Long Island, New York. But, it is correctly spelled "Rockville Centre".
When Clementine asks if she is ugly, Joel calls her "Tangerine" after the scene where she died her hair orange and he gave her this nickname. The film shows Joel's memories of Clementine working backwards from when she left to when they met, so Clementine would not have this nickname yet. However, because Joel is conscious of his memories (as well as manipulating them consciously at the time) and had heard Patrick in the previous scene calling Clementine by this nickname, it's possible that he calls her "Tangerine" for this reason.
The evening of his erasure procedure, Joel comes home with the prescription sleep medication and new pajamas he will need that night when the technicians plan to visit his apartment. As he comes home, he speaks to his neighbor in the lobby about Valentine's Day and "McRomance." During his memory erasure, he recalls this memory. However, since the memories slated for erasure have already been mapped in Lacuna's office, there is no way this particular memory could have been identified, mapped, and slated for erasure.
After deleting a memory, the camera pans from Joel in an overhead shot, moving then to Stan and Patrick. A bundle of wires can be seen on the floor in between the bed and table, one of which is obviously not plugged into anything. Director Michel Gondry pointed out this error in the DVD commentary of the film.