Continuity mistake: The bullets that Leary removes from the keychain at the banquet are in a different position than they were when he put them in.
Visible crew/equipment: As Leary is falling at the end, you can see the safety wire the stuntman is using as he falls backward.
Continuity mistake: When we see Frank peering out of Booth's window after the landlord calls the cops, it appears to be dark outdoors with only street lamps for light. The camera then cuts to Booth viewing Frank from across the street, but it appears that it is now daylight outside.
Continuity mistake: When Leary is assembling/loading his gun, the two bullets he slides out of the rabbit's foot appear to be .22 rim fire shells. The ones he slips into the gun appear to be either .32 or .38 center fire.
Continuity mistake: Clint Eastwood breaks a window in at John Malkovich's house, yet no broken glass is on the carpet.
Continuity mistake: As Frank goes down the glass elevator and stares at Mitch, the corpse's arms go from extended out to down toward his sides.
Factual error: When the motorcade arrives at the hotel, the Secret Service agent opens the door to his limousine and the President gets out. The door is then almost immediately shut. Proper procedure for the Secret Service is to hold the door open until the President is well away from the car, in case of an incident where the President has to be evacuated from the area quickly.
Factual error: The second time Frank speaks with Booth, the line is being traced. When Frank switches over, Okura tells him to keep Booth on the line until they get the ANI (automatic number identification). Assuming the secret service has access to the phone company's internal computer (since you have to have toll-free service or be a customer of an ANI outsourcing company to receive the subscription based info), ANI only says who the phone is registered to and since the call was being tapped they would've had the ANI information before the call was picked up. This still holds true if the ANI is faked by a well built PBX system (as said later).
Factual error: When Frank talks to Booth the first time on the phone and realizes he's at the phone outside, he runs to find him, only to find the handset swinging and the phone emitting the periodic screech that is heard after 1 minute of leaving a phone off hook. Seeing as how the apartment is three stories max, there's no way that sound would be playing by time he got there even if Booth somehow hung up first. Furthermore, even if it did take that long, the handset wouldn't still be swinging.
Revealing mistake: When Frank is hanging off the roof, there is a pipe from which he tries to cling to. The first time we see it there is a white line where it will break. The next time we see it (just before it breaks) the line has been coloured in.
Factual error: When Leary shoots at the president, the outer security ring of agents is missing. This is a standard protocol. There are 4 Secret Service agents immediately around him, and a secondary ring of 8-10 agents to handle things like a shooter.
Continuity mistake: When Clint Eastwood and Rene Russo go up to her hotel room after being in the bar, Russo was in a pantsuit and when she undressed, she was wearing a slip, which would not be appropriate when wearing pants.
Visible crew/equipment: The boom mic is reflected in the limo's windshield as Clint Eastwood jogs alongside it, after Leary's first phone call.
Factual error: Considering there are 229 people protecting the president, it is a pretty serious security breach that the sharp shooters at the end do not have night vision on their rifles.
Continuity mistake: In the breakroom scene where Horrigan is thought to have had a heart attack, the EMTs burst in and attempt to "revive" him and he shoves them away to both sides. The camera then shifts to an over-the-shoulder view, where one EMT says "I got a call there was a cardiac case." Both EMTs are present at the left and right edges of the screen. Seconds later, Frank says "Oh, I get it. Who's the wise guy?" and stands up. In that brief interval, the EMTs have vanished completely; when everyone else files out of the breakroom, laughing, they are nowhere to be seen.