Is it required by law to service powered pedestrian doors?

In short - YES 

Under the Machinery Directive (EU) or the Supply of Machinery Safety Regulations (UK) Powered Pedestrian Doors must be operated safely, safe for employees and safe for other users. The doors therefore need to be maintained in accordance with the guidance from the manufacturer, this should state what needs to be carried out and the frequency. However the law does not specifically state that the doors need to be serviced but failing to maintain them so they are safe in use would lead to a prosecution if there was an injury.

The law for servicing equipment is dependent on which country or devolved government you are in, however they all have similar content, here is an example for England and Wales:

Regulation 5 of the Workplace (Health, Safety and Welfare) Regulations 1992), state. Extract: "...... the equipment devices and systems to which this regulation applies shall be subject to a suitable system of maintenance."

Regulation 5 of the Provision & Use of Work Equipment Regulations 1998 (PUWER)

Extract A: "Every employer should ensure that work equipment is maintained in an efficient state, in efficient working order and in good repair."

Extract B: "Every employer should ensure that where any machinery has a maintenance log, the log is kept up to date."