The suspected arson attack happened at 3 Degrees Hair Design on Redforge Road in the early hours of Friday morning.
Anyone who was in the area of the Old Mallow Road/Dublin Hill between 2.15am and 3am and saw anything unusual is asked to come forward.
Gardaí are also urging taxi drivers who may have dash cam footage to make it available.
Anyone with any information can contact Mayfield Garda Station on 021 455 82 60
