Firstly, one should clarify: the concept of "legal tender" is a fairly modern one. In ancient times, gold and silver coins
had value because of their metal content, not because the government declared that they
had value. An unpopular emperor's coinage was (usually)
still made of
good silver and
good gold, and would be accepted as such.
The
Roman judgement of
damnatio memoriae was made against several emperors; though their names were obliterated from whatever monuments and documents could be found, coins were usually not defaced; they were either left alone or (perhaps more likely) melted down completely and the metal used to make
new coins. Very few surviving coins have been defaced; given that coinage of condemned emperors is not noticeably scarcer than that of other emperors, it can be concluded that damnatio did not significantly affect the coinage.
If one considers "acceptance by the government in payment of taxes" as a definition of "legal tender", then of course the government would have accepted old coins. Since the
fineness of the silver
denarius and the
weight of the gold
aureus both gradually declined over time, and the government itself was fully aware of this, they would have known that most older coins were of generally better
quality metal than newer coins and would therefore be worth more; this metal could then be recoined to the new debased
standard for instant profit to the government. For example, ten
denarii of
Caligula could be melted down to make about twelve early
denarii of
Nero, thirteen
denarii of
Hadrian or twenty
denarii of
Septimus Severus.
Finally, we must be cautious in making conclusions about "which coins circulated" at any given time when all we really have to go on are coins that survive to today; by definition, these are coins that were lost, buried, cached or otherwise ceased circulating.