Yes, the creature continues taking the damage while at 0 HP
Being at 0 hit points does not cleanse a creature of any conditions unless the text of that condition says otherwise. Furthermore, a creature at 0 hit points is still a creature; therefore, they will still take damage at the start of their turns, thereby suffering a failed death saving throw.
They will not necessarily die after three rounds, however.
Affected creatures can reroll the saving throw for the poison at the end of each of their turns, ending the poisoned condition on a success. A creature at 0 hit points still has turns, so they are still eligible to end the effect in this way.
Alternatively, if they roll a natural 20 on a death save, they will regain 1 hit point and become conscious again, causing their death saves to reset. A creature could feasibly survive until the end of the poisoned condition on this way.
They can also die in as little as one round, as a creature might roll a 1 on their first death saving throw, suffering two failed saves and then taking the damage from the mold for the final failed save.