Ogre mech is a very durable enemy, with very high health, armour and shielding. Like with other enemies in the game, its shielding will regenerate over time. In order to damage them, you need to fire at either its head or one of its shoulders.
The Pulse rifle fires in multiple bursts. Take cover when it begins to fire and use the gap in between each burst to retaliate.
The Ogre mech must charge its beam attack. The animation of this takes a few seconds and involves the mech throwing its arms up before firing each time. Use this time to get to the edge of a piece of cover, then as it fires move to the other edge of the cover. Often, the explosion's radius won't reach you.
When targeting one of the three highlighted areas stick with the Assault Rifle or Particle Beam as these have a decent range and allow you to choose your target area.
If you deal significant damage to the head of the mech, it will lose its eye and the ability to target. This will cause it to randomly fire its weapons around the room, possibly harming its allies.
If you do significant damage to an arm, it may lose that arm and one of its weapons. The mech's left arm houses the Pulse Rifle, and the right houses the Storm Launcher. Destroy whichever limb is giving you the most trouble.