When Arthur goes back to his house, he makes a cup of tea. When he drinks it, there is clearly milk in it although he never put in any. When it vibrates along the table, it is black again.
When Ford is scooting along with the trolley full of beer, he passes a red truck on the right side of the screen (Ford's left side); however, when the shot pans out, there are only red trucks on the left side of the screen (Ford's right).
When Mr. Prosser is talking to Arthur, Ford Prefect is in the background with a trolley full of beer before he has actually arrived.
When Arthur breaks free of the restraints at the dinner table, he still has an armrest attached to his right arm. After he hits the mice and steps back, it's gone.
When Trillian and Arthur are helping Zaphod out of Humma Kavula's temple, Trillian's arm switches from being by her side to holding up Zaphod between shots.
As the sperm whale is falling through the Magrathean atmosphere, the sound of a humpback whale is heard. However, this is not a normal sperm whale.
When Humma Kavula steps off of his dinner table, he is walking normally (taking a step) instead of the supposed mechanical movement of his many legs.
In a shot not long after Humma Kavula removes his glasses revealing only scars underneath, the corners of actual eyes are visible underneath the lenses.
When Ford is running to rescue Arthur and the gang as they are being shot at outside Humma Kavula's temple, Ford starts running to the rescue and is supposedly screaming but his mouth is clearly closed.
When Ford passes Arthur's caravan as he chases a group of Vogons with a towel, the cameraman and a few more crew members are reflected in the caravan.
A cameraman is reflected in the left side of the kettle when Arthur makes tea in his house at the beginning.
Arthur has a Babel fish inserted in his ear by Ford (in order to understand the Vogons), but minutes earlier, the people of Earth scream in understanding of the Vogons' message of their impending doom. (This same plot hole is present in just about all versions of the story).
When Arthur is speaking to Trillian (Zooey Deschanel) as she takes a shower, he briefly begins to address her by her real name (Zooey) then corrects himself.
When Marvin and Arthur are sitting by the inter-dimensional portals, Marvin mentions he is a "manically depressed robot." This not technically correct, as manic depression - aka bipolar disorder - is characterized by two extremes in behavior: sadness, lethargy, hopelessness, etc. (what Marvin displays); and mania - which is usually excessive activity (doing a lot of things at once) and sometimes happiness. Marvin did not really show mania.
Dent's "Best laid plans of mice and men" is a common misquote. It should be "The best-laid schemes o' mice an' men", and the full quote is "The best-laid schemes o' mice an' men, Gang aft a-gley," from the Robert Burns poem "To a Mouse," written in Scottish dialect. Translation: the best-laid schemes of mice and men often go wrong.