While the Romans—and the Greeks and Egyptians before them—knew of and used the milk of the poppy for medicinal purposes, we don't have any clear evidence for addiction to opium, the most likely candidate for abuse due to its prevalence in society.

Opium seems to have been a rather common ingredient in medical concoctions due to its soporific and pain reducing abilities. Pliny talks about some of these uses and even mentions that it can cause death when taken in sufficient quantities. He relates a certain Publius Licinius Caecina, a senator who reached the position of praetor, had committed suicide by taking too much opium because "an incurable malady [had] rendered existence quite intolerable to him." He goes on to say that many others do something similar. While this doesn't point to addiction, it does indicate that that Romans were aware of the dangers of taking too much opium.

Now, there is an instance that is is occasionally noted as a case of addiction. Marcus Aurelius, it is suggested, needed opium to sleep. There is a point in Galen's works where he talks about Marcus Aurelius using opium as part of theriac, a multiple ingredient prophylactic against poisons. Galen seems to say that Marcus Aurelius couldn't sleep after opium was removed from the mixture. But Scarborough argues that if one does a close reading, one can see that Galen was working in concert with Marcus Aurelius to modify the opium content, either up or down, as the emperor required. This is not the upward trend that would suggest addition. This, one of the few places were addiction might be suggested, therefore doesn't actually indicate addition.

One of the reasons why we lack clear evidence of addiction is that ancient medical writers, let alone authors in general, might not have know about the possibility of addiction nor known that it could be caused by substances like opium. The symptoms of addiction might have been considered a separate disease and not a result of the medicine that people were taking [n.b. this is my own speculation].

Now, as many hear know all too well, absence of evidence is not evidence of absence, but from the surviving texts that we have, it does not seem like there were any recorded cases of addiction.

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Sources:

Nencini, Paolo. "The Rules of Drug Taking: Wine and Poppy Derivatives in the Ancient World. VIII. Lack of Evidence of Opium Addiction." Substance Use & Misuse 32.11 (1997): 1581-1586.

Scarborough, J. "The Opium Poppy in Hellenistic and Roman Medicine." In R. Porter and M. Teich (eds.), Drugs and Narcotics in History. Cambridge University Press (1995): 4–23.

Veiga, Paula. "Opium: was it used as a recreational drug in ancient Egypt?", in: "Cultural and Linguistic Transition explored. Proceedings of the ATrA closing workshop Trieste, May 25-26, 2016", Ilaria Micheli (ed.), Trieste, EUT Edizioni Università di Trieste, 2017, pp. 199-215