They definitely aren't seen as often as energy blasts from the hands, but, going by the Dragon Ball Wiki, there are actually quite a number of characters throughout the whole franchise who have used them. Examples of characters who exhibit the ability are Tien, Tambourine, Piccolo, Garlic Jr, Nappa, Frieza, Cooler, Dr. Gero, Androids 16 and 19, Cell and Super Buu.
Namekians use it a lot: Piccolo, his father King Piccolo, Kami, and later Lord Slug.
Tien also used them in the original series. He shoots them out of his third eye.
An enraged Goku uses eye beams on a rock formation to bury Vegeta after the latter was killed by Frieza. It's the only time we see him do this.
Daizenshuu 7 claims that eye beams are different from the Ki Manipulation used in the series, and Dr. Gero's and Android 16's are different from that.
In The Sigh of Haruhi Suzumiya, the arc that depicts the filming of the fictional movie shown in that episode, the eye beams were for real, and nearly killed Kyon at least once. Even Nagato Yuki was caught unaware at how many types of Eye Beams Haruhi can imagine.
It should also be noted that the first manifested Mikuru Beam behaved like an actual cutting laser.
Emporio Ivankov has an attack that's similar to an eye beam called "Death Wink", where he blinks his eye to create a blast of air that hits his enemies or deflects projectiles.
Gild Tesoro, from the movie One Piece Film: Gold, uses his gold powers to create a giant golem that he pilots as a gigantesque mecha. Said mecha can shoot out laser beams from its eye.
Subverted in The Five Star Stories. The first time the colossal Jagd Mirage uses its eye lasers it ends up burning out its main cameras.
Afghanis-Tan has Meriken suddenly pull these out to attack Afganis after being bitten by her cat Al-Nyaida. If you haven't figured this out already, this is the comic's portrayal of 9-11 and the subsequent invasion of Afghanistan.
Stroheim also gets one that fires concentrated ultraviolet light. It proves to be extremely effective against creatures weak to sunlight, and backpack-mounted versions are later manufactured and distributed to foot soldiers.
In School Rumble, the freakishly huge head editor of where Harima's manga is being published has Eye Beams as well.
In Rebuild of Evangelion 2.0: You Can (Not) Advance, Unit 01 does this during the Zeruel battle when Shinji becomes pissed enough. Interestingly, it behaved like an actual beam rather than Zeruel's "deadly gaze" (i.e. its eyes twinkle, then the target is blow to hell and back with Creepy Crosses).
Speaking of Zeruel, other angels in Neon Genesis Evangelion have access to eye beams. Sachiel has the same "deadly gaze" that Zeruel has, complete with cross shaped blasts, and Israfel Kou and Otsu can both shoot simple eye beams.
The other two bigs, Big Fou and Big Duo also sport these. At the end of the series, Big Duo even gets its upgraded.
In Shinzo Mushrambo was able to instantly incinerate a mook with black fire just by glancing in his direction. He was of course evil and brainwashed at the time but that didn't change how powerful he was so chances are he could still do it.
In Digimon, the eyes are as good a place as any from which to launch one's beam attacks.
Digimon Adventure: Monzaemon has a fairly standard eye-beam attack that he uses while chasing Mimi and Palmon.
Digimon Adventure 02: Halsemon has two attacks like this. One, called Red Sun in the original and Eagle Eye in the dub, fires the standard red eye-beams. Another attack called Udjat Gaze that fires gold rings instead. Nefertimon can also fire these from the snake head on her headdress with Curse of Queen/Cat's Eye Beam.
Digimon Frontier: Special mention goes to Duskmon. His suit has eyes on it at various places, and his Deadly Gaze attack involves all of them focusing on you and firing death ray. Using this attack at far less than full strength can put Mons of higher levels in their place and a Digimon being able to take on a higher-leveled one is extremely rare. His evolved form Velgrmon also has an attack called Dark Vortex, which fires a beam from his center eye. If that weren't enough, the beam is also eye-patterned. Lobomon also has a light-elemental eye-beam attack called Strahl, though the attack goes uncalled in the dub.
In Digimon Fusion, Shakkoumon is able to fire eye-beams and rather impressively, he can match the power of Lucemon Falldown Mode's attacks with them. He also appears in Digimon Adventure 02.
In Digimon Universe: App Monsters, we get an Expy of Duskmon called Charismon. Instead of firing lasers directly from the eyes on his armor, he instead deploys eyeball-themed Attack Drones and lets those fire the beams. The eye on his helmet can also fire a beam attack.
This has proven to be a rather common antagonist ability in Digimon Adventure: (2020), but Eyesmon takes this to new levels. The individual bat-creatures that form its Scatter Mode being able to fire these from the shadows. While they prove to be too uncoordinated to turn it into a proper Beam Spam, Eyesmon's merged form is able to fire beams from all of the eyes on its head. Its Final form, Nidhoggmon is also able to fire beams from its regular eyes. As bizarre as this guy already is, Nidhoggmon's Breath Weapon is just it firing more eye-beams from its tongue and focusing them into a directed blast. Algomon Mega is no slouch either as it spawns dozens of eyes on the walls around it and fires beams from all of them.
Let's not forget Pegasus and Bakura's Shadow Game from Yu-Gi-Oh!: Ancient Egyptian Laser Beams!
Yudias Velgear from Yu Gi Oh Go Rush can fire green eye beams that are powerful enough to destroy phones and washing machines. He once even used them to repair a metal door after cutting a hole in it.
You would not expect this in a magical girl anime, but HappinessCharge Pretty Cure! pulls it off. With its main heroine.
Sgt. Frog: Taruru's main attack. Gyororo also has these.
Early in Pokémon: The Series, the move Foresight is rendered as red beams of light emitting from the eyes of the Pokemon using the move. It doesn't cause any damage, but it does dispel illusions and reveal Ghost Pokémon while making them vulnerable to Normal and Fighting attacks.
Bleach: In his Resurreccion, Kaname Tousen gains the ability to charge a lime-green Cero in front of both of his eyes, which then fuse together before being fired at the enemy. Appropriately, this attack is named "La Mirada" (Spanish for "The Gaze").
Patlabor: The TV Series: The Phantom, the Labor that uses the aforementioned Energy Weapons, has a creepy, skull-like face & fires the beams out of the skull's "eyes", but the main camera is actually located in the "mouth".
on a rock formation to bury Vegeta after the latter was killed by Frieza. It's the only time we see him do this.
