Why does Palpatine's voice change when he reveals he is a Sith? Does the prolonged use of the Dark Side during his fight with Windu cause his voice to change, like when his face is scarred?
He has been seriously burnt by the counter to his lightning. He started to speak differently just after that, with Anakin. It may just be the consequences of the burn.
It may also have been a trick to appear more severely injured by the "treason" of the Jedi in front of the Council, and he kept that voice after that for the darker effect.
I don't buy the 'scarred' theory. To my recollection, in the scenes of The Phantom Menace where Sidious speaks to Nute Gunray, he speaks in his 'Sith' voice.
I think the darker voice is Palpatine's actual voice; the 'normal' voice he uses around others is a practiced performance, just like the rest of his politician persona.
Palpatine/Sidious was masking his true corrupted, disfigured appearance/voice all along, with a Sith spell that was broken once Windu deflected his lightning back to him.
The dark side of the force has a tendency to disfigure its users in a phenomena known as "dark side corruption".
"Will I eventually be physically transformed?" "Into some aged, pale-skinned, raspy-voiced, yellow-eyed monster, you mean. Such as the one you see before you. Surely you are acquainted with the lore: King Ommin of Onderon, Darths Sion and Nihilus. But whether it will happen to you, I can't say. Know this, though, Sidious, that the power of the dark side does not debilitate the practitioner so much as it debilitates those who lack it. The power of the dark side is an illness no true Sith would ever wish to be cured of."
―Darths Sidious and Plagueis discuss the physical signs of dark side immersion, in Darth Plagueis (novel)
The malevolent power of the Dark Side takes a toll on the body, causing the disfigurement, gaunt yellow eyes and distorted voice.
Palpatine had been using an alchemical Sith spell to mask his true appearance, which was broken once Windu deflected his lightning back to him. In other words he always looked/sounded like that and only masked his appearance, with Sith alchemy, until such a time when it was no longer necessary (Mace Windu's death). In fact, he uses his drastic change of appearance to his advantage, blaming the Jedi, whenever he is making his speech to the senate.
01:31:18 - Palpatine: The attempt on my life...
01:31:21 - Palpatine: has left me scarred and deformed.
Star Wars: Episode III - Revenge of the Sith (2005)
Others have suggested that the lightning damaged him, but there are plenty of examples before that where he reveals his true voice (including Episodes I, II and III and even the Clone Wars series). Indeed, even in the scene where Windu confronts Palpatine he lets his true voice slip:
Note: this information is canonical only insofar as C-canon (Continuity canon) allows. Moreover, the Disney acquisition of Star Wars in 2012 upset 30 years worth of canon, doing away with a vast array of canonical information, which complicates matters for my answer's canonical validity.
In short: if you consider pre-2014/Holocron continuity database's idea of what is and isn't canon to be legitimate, then my answer is canonical (C-canon) for you. If, on the other hand, you consider post-2014, Disney-established Lucasfilm Story Group's judgement of what is and isn't canon to be legitimate, then my answer relies on non-canonical information.