Goofs

Factual errors

When Dr. Erin Mears is in Minneapolis racing to isolate the infected IMM executive from the bus, she can be seen wearing an N95 mask with an exhaust vent, about to put a similar one on the victim to limit his infectiousness to others. Vented N95 masks have a one-way valve that lets exhaled breath bypass the filtering for easier exhalation. She should be using a non-vented N95 mask on the infected patient.

Character error

When Alan Krumwiede is explaining what an R0 of 2 means on television, he claims that the number of cases daily will progress 2, 4, 16, 256, 65536 and so on. This is incorrect - this is the quadratic series, and would mean that by the fourth day each person is infecting 256 others, and by the sixth day 4,294,967,296 would have fallen ill. The correct progression would be the geometric; i.e. 2, 4, 8, 16, 32... He later claims that it would take a month for a billion people to fall ill - this number is clearly based on the geometric number series, not the quadratic.

Continuity

When Dr. Cheever leaves Roger's house after vaccinating Roger's son, the first shot shows his car parked on the wrong side of the street, with a huge pile of trash only 3 feet in front of the car. In the next shot from Cheever's point of view, the pile of trash is on the other side of the street, across the street from his car, instead of in front of it.

Errors in geography

During the expert meetings at the CDC, the map has New Delhi, which is in the far north of India, marked almost at the center of the country.
During the preliminary press conference at the CDC, Boston is listed as one of the first infected cities. Later on, a map is shown of infected regions around the world. It indicates that the only regions of Massachusetts infected are those regions west of the Connecticut River. Boston is approximately 100 miles east of the Connecticut River.
When Dr. Cheever is describing the mortality rate of the contagion (one in twelve) the North American map Montreal marked as Toronto.
Durban, South Africa, is listed as one of the first infected cities. Later on, a map is shown of infected regions around the world, it shows South Africa's west coast as entirely affected, but leaves out the east coast - where Durban is.

Factual errors

The disease in the film is highly lethal, affects a very large number of people and has a short incubation period. In reality an infectious disease must have a long incubation period and less lethality than in the film to facilitate a sustained transmission. The real case makes tracking much more difficult, which is a central part of the film, therefore the filmmakers had to bend the facts a bit.
When the 'virus structure' is explained by Dr. Cheever, just a protein structure is shown. What is termed as a receptor in green is just a phenylalanine (amino acid), only a small unit of a protein. Viruses have several proteins and proteins are built of amino acids, so what is shown can't be a virus at all.
At the end of the movie they are giving the vaccine which is an inhaled vaccine and telling the patients to breathe in through their nose. In reality, one should NOT breathe in while getting an intranasal vaccine as the attenuated virus must replicated in the warm nasal passage before spreading to the rest of the airways so your body can make antibodies against the virus.
In several scenes, Rear Admiral Lyle Haggerty's (Bryan Cranston) military decorations are shown. However, it appears that the decoration in the least prestigious place is the U.S. Army's "Distinguished Service Medal". Although it is not impossible for a Naval officer to have this decoration, it IS impossible for it to be in this position.

Incorrectly regarded as goofs

When Li Fai, the Hong Kong patient, exits the elevator on his way home, the two red Chinese characters on the wall means "the 10th floor", while the display of the elevator says it's the 9th. However, this is likely not a goof. Floor numbering has become quite confused in Hong Kong due to a combination of British numbering (ground floor -> 1st -> 2nd -> 3rd) and the Chinese system which counts the ground floor as the 1st. Also, due to other traditions in regard to the number 4, that number may be skipped when numbering the floors.
Even though the population of Minneapolis, Minnesota is technically just over 300,000, the 3.3 million number as annotated in the film refers to the total metropolitan area and not just the city.

Revealing mistakes

Right after Dr. Ellis Cheever announces that approximately 1 in 12 people will contract the disease, we see a montage of abandoned places from around the world. One of locations is a deserted office in what we would assume to be a deserted city. Unfortunately, several cars can be seen driving through the window on the right side of the office, depicting what one would see in a typical, bustling city.

Spoilers

The goof item below may give away important plot points.

Continuity

When Dr Leonora Orantes discovers the identity of Patient Zero and her liaison Sun Feng stands by, Feng's polo shirt starts the scene inside-out. As the two start for the car, his shirt is worn appropriately. Then in the van it is inside-out again and then correct, repeating throughout that sequence of events.