Why would you want to film the inside of an airport?

BTW The answer to your question is not going to be a simple yes or no. It depends on what you are actually filming (planes, specific individuals, personnel, etc.), how you film, where you film (e.g. security, immigration, etc.) and for what purposes you film. There have been several professional crews that have filmed at LHR (and other airports) for commercial purposes and you can be pretty sure that all of them did so with the explicit authorisation of the airport and the authorities involved.