We can say that Antony's tone moves between two poles - irony and sincerity.
Sarcasm and irony play against deeply felt remorse for the death of Caesar. Antony puts his grief on display to help work up the crowd. He invites the people to identity with him and not with Brutus and Cassius. In doing so, Antony suggests that the muderers have brought grief to Rome; to the Romans in the crowd.
The tone does, indeed, shift from one of apparent sincerity and love as he addresses the plebians and declares that he comes to speak of his friend. Surreptitiously, however, Antony injects his ironic remark that "Brutus is an honorable man" and then creates doubt of this honor as he provides arguments against the conspirators' claims against Caesar. Then, Antony assumes a very dramatic tone as he points to where Cassius's and Brutus's daggers went through Caesar, In an abrupt change of tone, Antony becomes dramatic and, in the rhetorical technique of apostrophe, he calls upon the gods as he turns the direction of his speech upon the conspirators' as "traitors" and their act as "bloody treason."
I agree with post #2. The shift in tone is due to his appeal to the Roman crowd, who he wants to turn against Brutus. His opening, that he wants to bury Caesar, not to praise him, is revealed as completely false as the speech develops. His characterization of Brutus is deeply sarcastic as he portrays him not as honorable, but in fact as a traitor.
This would be a good question to pose in the Q&A section of eNotes, where you are likely to get a more detailed response. In general, the tone of Antony's narration is highly ironic, and it becomes more and more ironic each time Antony claims that Brutus is an honorable man. Each time Antony makes that pronouncement, he is twisting the rhetorical knife a little deeper into Brutus's back.