What effect is achieved by having both Antony and Octavius speak well of Brutus at his death in Julius Caesar?

The venerating of Brutus serves to foreshadow trouble between Antony and Octavius, who are both trying to bring about peace and gain more power for themselves.

When you are the victor, you can afford to be eloquent.  Antony and Octavius both had reasons for posthumously pardoning Brutus.  They want an end to the civil war, highlighting the fact that they are the ones in charge.  The best way to bring about peace quickly is to take Brutus’s soldiers into their own armies.  To do this smoothly, they want to acknowledge what Brutus was fighting for.

Antony begins by saying that Brutus was “the noblest Roman of them all” (Act 5, Scene 5).  He goes on to explain that Brutus did what he did not out of ambition or anger, but because he genuinely believed it was good for Rome.

All the conspirators save only he
Did that they did in envy of great Caesar;
He only, in a general honest thought
And common good to all, made one of them. (Act 5, Scene 5)

Fine sentiments of course.  He is saying that Brutus died an honorable death, and therefore his soldiers can maintain their honor.  Antony wants those soldiers to like him.  He has been maintaining a precarious rule with Octavius and Lepidus, but would rather be the sole ruler of Rome.

Octavius feels the same way.  As much as Antony is loved by Roman soldiers, he cannot match Octavius for cunning and guile.  Octavius acknowledges Antony’s words, and goes one step further.

According to his virtue let us use him,
With all respect and rites of burial.
Within my tent his bones to-night shall lie,
Most like a soldier, order'd honourably. (Act 5, Scene 5)

You have to remember that Brutus was responsible for killing Octavius’s “father,” Julius Caesar.  He has every right to want revenge.  Still, by being gracious and taking charge of the body, he is saying that he harbors no ill will.  (Of course, rumor has it that he actually detached Brutus’s head and tried to send it back to Rome.  Clearly there was ill will.)

Note also that Octavius has already absorbed all of Brutus’s soldiers into his own army, when he said “All that served Brutus, I will entertain them.”  He is making a play to make himself stronger, so that he can eventually oust Antony and be the sole ruler of Rome.  It takes some time, but he does it.

This scene perfectly describes the politics of Rome.  We fought a war, now we are all on the same side.  It also foreshadows the trouble between Antony and Octavius.  The two are hardly best friends, and share power only grudgingly.  Each is biding his time until he can oust the other.  This will not be Rome's last civil war.