The plane was due to fly to the Spanish resort of Alicante when the incident took place on Thursday morning.
Its passengers and five crew members were quickly evacuated from the aircraft.
Footage showed people sliding off the plane before being helped to their feet, with others being carried in people's arms.
The runway was closed for one hour during the incident, Exeter Airport said.
A replacement aircraft departed on Thursday afternoon and was expected to arrive in Alicante at 4pm local time.
Flybe apologised to passengers for "any inconvenience experienced".
In a statement, the airline said: "All passengers have been issued letters upon boarding in which they have been informed that they will be refunded the full cost of their flight in addition to EU261 compensation.
"The safety of its passengers and crew is the airline's number one priority and Flybe apologises for any inconvenience experienced."
