The short answer is no, it would not.
The reasons why are multiple. I will keep the answers fairly non-technical - please forgive the loss of precision and technical accuracy in my answers as a result. I used to work in a research and development center where a large number of the scientists used lasers as part of their every day research, so I have gained a certain amount of familiarity with the physics involved.
First off, most reflective surfaces are not 100% reflective. As a result there is a slight energy loss as some of the light will pass through the surface, or be absorbed by it. This is due to the fact that a laser beam transmits some of the beam's energy in the form of heat to the surfaces it strikes. This causes the laser beam to lose a little bit of energy each time it bounces.
As a laser passes through an atmosphere, it excites the air particles it passes by and through, transferring energy from the laser to those particles in the form of additional heat loss. Finally, even though a laser beam is focused into a tight beam, it still radiates light - otherwise you would not be able to see it from the side. This represents another loss to the energy of the laser beam.
Eventually, through these multiple methods of energy loss, the laser beam would lose cohesion and dissipate. Even if you had some theoretically perfect reflectors in perfect alignment, enclosed in a container which was holding a theoretically perfect vacuum, the radiant energy given off by the laser would be absorbed by the container and reflectors as heat and still eventually cause the laser beam to lose cohesion and dissipate.