Sort of. If you went to a theater performance in Ancient Rome, you wouldn't expect to see an actual death. However, public executions at the games took on theatrical aspects. As early as the 30s BCE, a certain bandit leader and/or insurrectionary's execution at Rome (described by Strabo) referenced and dramatized the geography of his crime:

And recently, in my own time, a certain Selurus, called the "son of Aetna," was sent up to Rome because he had put himself at the head of an army and for a long time had overrun the regions round about Aetna with frequent raids; I saw him torn to pieces by wild beasts at an appointed combat of gladiators in the Forum; for he was placed on a lofty scaffold, as though on Aetna, and the scaffold was made suddenly to break up and collapse, and he himself was carried down with it into cages of wild-beasts — fragile cages that had been prepared beneath the scaffold for that purpose.

Tacitus describes the punishment of a large group of Christians in the following way. The reference to condemned people being killed in animal skins by dogs suggests that the spectacle was intended to evoke a hunt:

Mockery of every sort was added to their deaths. Covered with the skins of beasts, they were torn by dogs and perished, or were nailed to crosses, or were doomed to the flames and burnt, to serve as a nightly illumination, when daylight had expired.

Martial describes a forced punitive reenactment of the story of Mucius Scaevola (the real Scaevola supposedly held his right hand in fire to make a point about bravery and the triviality of the body):

If Mucius, whom you saw in the amphitheatre one morning recently putting his hand in the fire, seems to you stoical, unflinching, and strong, you have the intelligence of the mob from Abdera...

Most often, though, elaborate theatrical executions involved dressing a criminal up as a Greek mythological figure, and giving them a punishment suitable for the figure's story. Tertullian writes the following:

But you really are still more religious in the amphitheatre, where over human blood, over the polluting stain of capital punishment, your gods dance, supplying plots and themes for criminals-unless it is that criminals often adopt the roles of your deities. We have seen at one time or another Attis, that god from Pessinus, being castrated, and a man who was being burnt alive had taken on the role of Hercules.

Likewise Martial describes someone killed in the persona of Orpheus, whose death by a crowd of beasts was an ironic divergence from Orpheus' mythical ability to charm wild animals. He also references "Daedalus" being killed by bears (which has no relation to Daedalus' myth) and "Pasiphae" being penetrated by a bull in an amphitheater.

The extent of the possible play-acting is not clearly described. While theatrical plays were often performed in the same spaces or events as the bloodier forms of entertainment, they were a pretty different genre, and scholars generally interpret these killings as spectacles combining well-known stories with execution, not as the insertion of criminals into full plays with scripts. After all, being an actor, then as now, was a skill people learned and practiced; and the Roman plays that survive would not lend themselves very well to including execution. (For example, Hercules as a character is well-suited to this kind of execution because he is a well-known character who dies dramatically, but in Hercules furens he doesn't die and in Hercules Oetaeus he comes back as a god and has more lines after he dies.)

There were also a number of mock naval (and sometimes land) battles in which a crowd of criminals could be killed at once, often reenacting a famous real battle.

There may be a part in many of us that is both fascinated and horrified by stories of the extent of Roman state violence, and/or that is influenced by the Christian tradition of glorifying and fetishizing the experience of those who suffered under that violence. So I want to close by emphasizing that we're talking about the torture and death of real people, and that these theatrical executions were a part of the humiliation and mockery of condemned criminals and part of Rome's imperial machine.

Bibliography

Coleman, K. M. 1990. "Fatal Charades: Roman Executions Staged as Mythological Enactments". The Journal of Roman Studies 80, 44–73.

Epplet, C. 2014. "Spectacular Executions in the Roman World", in A Companion to Sport and Spectacle in Greek and Roman Antiquity.