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So yes, while lasers will exert radiation pressure upon solar sails (see @PearsonArtPhoto' answer for more information), they will also exert a force equal in magnitude and opposite in direction on the vehicle's body, as per Newton's third law of motion. And because the lasers are a part of the same system (not two separate bodies, as quoted in Newton's third law of motion - they're attached to the vehicle and it to the solar sails, possibly by some long tether), this force on the solar sail will be canceled out by the equal and opposite force on the lasers that are emitting the photons. It would stretch the tethers the sails are connected to with the spaceship, but that's about it.

Edit to add: I see in the comments that my and @PearsonArtPhoto's answers have caused some concerns over their accuracy, due to supposed "free lunch" concept of pushing on solar sails using an onboard generated laser beam. So what follows is a bit of background on the actually proposed mode of propulsion, as found in scientific papers. If anyone can find a proposed solution that does this in a closed-loop system, and propels a spaceship in any direction by focusing a laser beam that is aboard the spaceship on the solar sails attached to the same spaceship, I'll be extremely interested to read the paper.

So yes, while lasers will exert radiation pressure upon solar sails (see @PearsonArtPhoto' answer for more information), they will also exert a force equal in magnitude and opposite in direction on the vehicle's body, as per Newton's third law of motion. And because the lasers are a part of the same system (not two separate bodies, as quoted in Newton's third law of motion - they're attached to the vehicle and it to the solar sails, possibly by some long tether), this force on the solar sail will be canceled out by the equal and opposite force on the lasers that are emitting the photons. It would stretch the tethers the sails are connected to with the spaceship, but that's about it.

Edit to add: I see in the comments that my and @PearsonArtPhoto's answers have caused some concerns over their accuracy, due to supposed "free lunch" concept of pushing on solar sails using an onboard generated laser beam. So what follows is a bit of background on the actually proposed mode of propulsion, as found in scientific papers. If anyone can find a proposed solution that does this in a closed-loop system, and propels a spaceship in any direction by focusing a laser beam that is aboard the spaceship on the solar sails attached to the same spaceship, I'll be extremely interested to read the paper.

Edit to add: I see in the comments that my and @PearsonArtPhoto's answers have caused some concerns over their accuracy, due to supposed "free lunch" concept of pushing on solar sails using an onboard generated laser beam. So what follows is a bit of background on the actually proposed mode of propulsion, as found in scientific papers. If anyone can find a proposed solution that does this in a closed-loop system, and propels a spaceship in any direction by focusing a laser beam that is aboard the spaceship on the solar sails attached to the same spaceship, I'll be extremely interested to read the paper.

First, from Wikipedia on Beam-powered propulsion - Direct impulse:

The beam has to have a large diameter so that only a small portion of the beam misses the sail due to diffraction and the laser or microwave antenna has to have a good pointing stability so that the craft can tilt its sails fast enough to follow the center of the beam. This gets more important when going from interplanetary travel to interstellar travel, and when going from a fly-by mission, to a landing mission, to a return mission. The laser or the microwave sender would probably be a large phased array of small devices, which get their energy directly from solar radiation. The size of the array obsoletes any lens or mirror.

Right. Not really directly dismissing the idea of carrying the source of the laser beam aboard the spacecraft, but it is suggested the beam source would have to be stationary by implying the greater radii (due to angular resolution) the more distant the spaceship is to the energy beam source. Let's take this a step farther and quote actual scientific papers, not only Wikipedia (which is fine in most cases, but one can't trust conclusions there with any degree of certainty, unless also absolutely certain it doesn't change implied meaning of its quoted sources);

From Advanced Solar and Laser-pushed Lightsail Concepts, Final Report, May 31, 1999 - NASA Institute for Advanced Concepts (PDF):

1.11 Beamed-energy Propulsion (pages 4-6)

An alternative solution to the problem of the mass ratio required for high velocity flight is to use beamed energy. In beamed-energy propulsion, the energy source is left stationary, and the probe is pushed at a distance. Since the propulsion system does not move, the weight of the energy source is not critical, and fuel does not have to be carried.

An example of the beamed-energy propulsion is the photon-pushed sail. Since a photon has momentum, a photon beam can “push” a reflective sail. In practical terms, the force produced by reflecting a light beam is 6.7 newtons per gigawatt of light reflected. This force comes with no expenditure of fuel whatsoever. Thus, it is extremely favorable for high delta-V missions.

So there we have it. Such systems describe a stationary beam source, and a moving spaceship pushing against the momentum force reflected photons exert upon the solar sail.

I have also watched the first laboratory tests performed on feasibility of using this momentum of the photons targeted at an extremely light solar sail, but sadly don't exactly remember where this study was conducted, or who the publisher was (it was a fair bit ago, probably around 10 years?). Still, the test system was assembled in a vacuum chamber with the high-power laser stationary, and a small (roughly 1 square inch in size) solar sail held in place by an extremely sensitive lever that didn't impart movement of the test piece of the solar sail in the direction of the light beam. The forces measured were extremely tiny, but the concept worked and the method proposed was confirmed. Again - by using a stationary laser beam source!

Hope this clarifies things a bit.

I'm not trying to be smart, or mock your proposal in any way whatsoever. I just genuinely love Baron Münchhausen's stories since I was a kid, and this one, where Münchhausen allegedly pulled himself and the horse on which he was sitting out of a swamp by his own hair, is as good of analogy as I can currently think of, and it taught me the very same dynamics of a closed system that I'm describing here when I was a kid and didn't even hear of Newton before.

Now, if the laser was stationed on another body, such propulsion would be quite possible. It could also be used to constantly change direction (spin on its axis), even if it was part of the same body, but it wouldn't work as a long term propulsion in any desired direction. Sadly, there's no free lunch here, not even with photons.

I'm not trying to be smart, or mock your proposal in any way whatsoever. I just genuinely love Baron Münchhausen's stories since I was a kid, and this one, where Münchhausen allegedly pulled himself and the horse on which he was sitting out of a swamp by his own hair, is as good of analogy as I can currently think of, and it taught me the very same dynamics of a closed system that I'm describing here when I was a kid and didn't yet even hear of Newton until then.

Now, if the laser was stationed on another body, such propulsion would be quite possible. It could also be used to constantly change direction (spin on its axis), even if it was part of the same body, but it wouldn't work as a long term propulsion in any desired direction. Not by targeting the photon emitting lasers on a sail attached to the same vehicle. If would work without a sail though. Sadly, there's no free lunch here, not even with photons.