What happened to the coins in ancient Greece and Rome when monarchs changed?

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Ancient Greece and the Hellenistic period

Coins in Classical Greece did not bear the image of rulers so there was no 'personal' motive to recall coins issued during the time of previous rulers. Coins featured images of gods, goddesses, animals and objects. The most widely distributed coinage in Classical Greece was that of Athens, which was

uniform to the point of monotony - an uninspired repetition of the old formulas, and often carelessly executed at that.

As for the process by which old coins were replaced, there is this example: during the latter stages of the Peloponnesian War, Athens ran short of silver and issued bronze coins (around 406/405 BC). Later (when exactly is disputed, but possibly as early as 404 BC), the Athenians reverted to silver. J.H. Kroll, citing Adalberto Giovannini, (pdf) says this was done thus:

the withdrawal of the bronze through an exchange for silver may have been a gradual process that began in 404 and only ended with the herald’s declaration of demonetization.

Aristophones refers to these changes in coinage in two of his plays, The Frogs (405 BC) and Ecclesiazusae (392 BC).

If your primary interest is monarchs, only the Spartans among the major city states had kings (a dual monarchy) in the classical period, during which time they did not issue coins.

Despite the profusion of kings during the Hellenistic period (as noted by Ginasius in a comment below) following the death of Alexander the Great, rulers usually left old coins in circulation unless they had economic reasons for withdrawing them. For example, as the finances of the Seleucid rulers worsened over time, the currency was debased. Thus,

As lower value coins were put into the system, earlier Seleucid and foreign coins of higher intrinsic worth were taken out.

Some coins of Ptolemy I were also restruck in Egypt by Ptolemy II, but this was not about erasing all traces of his father and predecessor; the restriking included foreign coins as well as some Ptolemy II trichrysa from earlier in his reign. Ptolemy II's aim was instead to create a new coinage which would have a monopoly, meaning all foreign coins had to be exchanged (a policy which his father had started).

One example of 'erasing' a previous ruler can be found with the Indo-Greek Satavahana ruler Gautamiputra Satakarni who (in the 1st or 2nd century AD),

vanquished the Western Satrap ruler Nahapana.... This victory is known from the fact that Gautamiputra Satakarni restruck many of Nahapana's coins,...

Ancient Rome

Coins in Ancient Rome usually remained in circulation even after an emperor was deposed or died. Changes usually came about due to changes in the monetary system and / or because of economic circumstances, which more often than not meant debasing.

One case where coins of a specific emperor were recalled appears to be Pescennius Niger, emperor from April 193 to May 194. According to Roman Numismatics

The scarcity of Pescennius Niger's coinage today belies the fact that it was struck on a monumental scale, and we can only assume that after his defeat at the hands of Septimius Severus in AD 194 his coins were meticulously recalled and melted.

Although Pescenius Niger coins are rare today, they were struck in large numbers and at

a standard markedly inferior to that of any previous Roman denarii, below even those of Commodus’ last three years.

Source: A. R. Birley, Septimius Severus

As Niger only ever controlled parts of the eastern empire, his coins were all struck there, mostly in Antioch. After his defeat by Septimius Severus, Niger's family were executed, his lands confiscated and his supporters punished. Further,

portraits honoring Pescenius Niger were destroyed as a consequence of his proclamation as a hostis. [enemy of the state]

There was thus a concerted effort at removing all traces of Niger, who had been a rival to Severus; as coins were a valuable propaganda tool for emperors and (with the emperor's head on every coin) a symbol of the unity of the empire, Niger's coins would have been an obvious target for destruction. Severus was quick to mint coins with his head, some of which may have been exchanged for Niger coins handed in.

Medieval and Early Modern Examples

In England, after the Norman conquest, William I

issued huge numbers of silver pennies and operated more than seventy mints as the coins of former Anglo-Saxon monarchs poured into the melting pot to emerge from the striker’s anvil bearing the new king’s portrait.

Source: C. H. Perkins (ed), England's Striking History

Later, after the English Civil War (1642-51), coins bearing the head of Charles I (who, of course, lost his head in 1649) were replaced. The restoration in 1660 led to coins minted during the Commonwealth being