Continuity mistake: In 'Consequences' Buffy says to Faith 'Less that 24 hours ago you killed a man'. In 'Bad Girls', Faith kills Finch in the alley on the first night. One night later Buffy and Angel are attacking Belthazor, the night after that is the night when Buffy dreams about drowning, so the night when Buffy says this is actually about 72 hours after the murder. (00:11:25)
Continuity mistake: When the detective is interviewing the woman in the alley, he mentions that the murder took place "last night". In "Bad Girls", we see Buffy visiting Faith the day after the murder, and that night is presumably when she has the dream about drowning, so by the time the detective is asking this question it is actually two nights after the murder, not one.
Other mistake: When the detective is interviewing the woman in the alley he says "So you heard the man scream about what time?", but Finch didn't scream as he died.
Chosen answer: He had larangitis and because of the shooting specifications they had to have him shoot the scene with a bad throat, they then re-recorded his lines and dubbed them. At various points during the scene and indeed the episode you can see him touch his throat or rub it.