Why was it a mistake for Brutus to allow Antony to have the last word at Caesar’s funeral in Julius Caesar?

It was a mistake to allow Antony to speak after Brutus because Antony was a more persuasive speaker.

Brutus was convinced that Mark Antony was harmless.  He believed that Caesar was the only one that had the power, and as long as Caesar was dead Antony would leave them alone.  He does not see Antony as any kind of threat.

Brutus does not want to kill anyone other than Caesar because he fears that the conspirators will be seen as butchers and not tyrant-killers. 

We shall be call'd purgers, not murderers.
And for Mark Antony, think not of him;
For he can do no more than Caesar's arm
When Caesar's head is off. (Act 2, Scene 1)

Cassius tells Brutus that he fears Antony, because he loved Caesar and because he is a “shrewd contriver.”  Brutus should have listened to Cassius, but he never listened to Cassius.  He was in charge, and his opinion was what mattered.

Brutus’s idealism got the better of him when Antony asked to speak at Caesar’s funeral.  Once again, Cassius tries to talk Brutus out of it. 

You know not what you do: do not consent
That Antony speak in his funeral:
Know you how much the people may be moved
By that which he will utter? (Act 3, Scene 1)

Cassius apparently knows what a good speaker Antony is.  Brutus thinks that Antony will be harmless because he is speaking after Brutus.  However, he thinks too much of his abilities.  Antony is a much better speaker than Brutus. 

Brutus’s speech is all logic and excuses, but Antony’s seems to be from the heart.  He tells the people that he is not there to make excuses for Caesar, but that he loved him.  He also holds up Caesar’s mantle and points to each place a conspirator stabbed, naming each of them.  Finally, he shows the people Caesar’s will and tells them that Caesar left his house and gardens to them and also left money to each of them.

Antony’s theatrics win the day.  He transforms Brutus’s calm crowd into an angry mob, out for the conspirators’ blood.  Antony is such a good speaker, in fact, that he convinces the people that they should follow him instead of Brutus and the conspirators are forced to flee.