The door opened as the train approached Terminal 4 Heathrow, where passengers were taken off the service.
The Rail Maritime and Transport (RMT) union said it was the fifth time it had happened in recent weeks.
Transport for London (TfL) said it had carried out checks which showed it was an "isolated incident".
The RMT said some drivers were refusing to work on Sunday due to safety fears.
The ASLEF union also said some drivers were refusing to operate the Tube trains.
TfL said it had met with trade unions to give them full assurance there were no wider safety issues.
The Piccadilly line is operating with delays and some suspensions.
Pat Hansberry, London Underground operations director, said: "Following an isolated problem with one train door we carried out a full inspection to confirm there wasn't a wider problem, which resulted in some cancellations."
The RMT has previously raised questions about the reliability of the Piccadilly Line trains, which it describes as an "ageing fleet".
A spokesman for the union said: "There are serious questions over the safety of the ageing fleet on the Piccadilly Line in terms of braking systems, wheel flats and brake operation that RMT has been raising for well over a year now.
"There are clearly major safety issues which must not be ignored any longer".
A train door opened on a moving Piccadilly Line train on Saturday morning, prompting an inspection of all the trains on the Piccadilly Line.