Specifically, besides granting Sauron spectator mode of the Battle at the Black Gate, does it grant any combat bonuses to his troops or penalties to Aragorn's army?
Feel free to answer this question with a general explanation of the Eye's capabilities.
There's little evidence to suggest that the Eye itself confers any combat advantage; there's only one instance I can think of to indicate any special powers of the Eye, but that case is muddied somewhat by the presence of the Ring (emphasis mine):
[W]ith every step towards the gates of Mordor Frodo felt the Ring on its chain about his neck grow more burdensome. He was now beginning to feel it as an actual weight dragging him earthwards. But far more he was troubled by the Eye: so he called it to himself. It was that more than the drag of the Ring that made him cower and stoop as he walked. The Eye: that horrible growing sense of a hostile will that strove with great power to pierce all shadows of cloud, and earth, and flesh, and to see you: to pin you under its deadly gaze, naked, immovable. So thin, so frail and thin, the veils were become that still warded it off. Frodo knew just where the present habitation and heart of that will now was: as certainly as a man can tell the direction of the sun with his eyes shut. He was facing it, and its potency beat upon his brow.
The Two Towers Book IV Chapter 2: "The Passage of the Marshes"
It seems unlikely that someone like Aragorn, not carrying the Ring, would feel that effect; it's notable that the next paragraph implies that Sam doesn't (emphasis mine):
Gollum probably felt something of the same sort. But what went on in his wretched heart between the pressure of the Eye, and the lust of the Ring that was so near, and his grovelling promise made half in the fear of cold iron, the hobbits did not guess: Frodo gave no thought to it. Sam's mind was occupied mostly with his master hardly noticing the dark cloud that had fallen on his own heart.
The Two Towers Book IV Chapter 2: "The Passage of the Marshes"
It's worth reiterating that the Eye is not Sauron himself; Sauron most assuredly does have some ability to buff his forces and it's possible, but by no means explicit, that he can also debuff his enemies (emphasis mine):
But the Nazgûl turned and fled, and vanished into Mordor’s shadows, hearing a sudden terrible call out of the Dark Tower; and even at that moment all the hosts of Mordor trembled, doubt clutched their hearts, their laughter failed, their hands shook and their limbs were loosed. The Power that drove them on and filled them with hate and fury was wavering, its will was removed from them; and now looking in the eyes of their enemies they saw a deadly light and were afraid.
Then all the Captains of the West cried aloud, for their hearts were filled with a new hope in the midst of darkness.
Return of the King Book VI Chapter 4: "The Field of Cormallen"
But those powers are related to his powers as an Angelic spirit, not any special ability of his giant flaming Eye.
