The published Metin2 documentation does not provide a separate rule for what happens to Dragon Stone time after logout. It therefore does not directly support a definitive yes-or-no answer about offline time.
What the documentation does establish is that every Dragon Stone has a 24-hour usage duration and that this time runs only while the stone's corresponding set is active. Gameforge's Metin2 Alchemy Guide states this rule directly, and the Metin2 Wiki describes the same activation-based timer.
Active and inactive sets
A character can equip two Dragon Stone sets, but both cannot be active simultaneously. Activating a set starts the timers for the stones in that set. Under the documented active-set-only rule, stones in an inactive set do not consume their remaining usage duration.
Neither source explicitly states whether logging out deactivates an active set or otherwise changes its timer state. Confirming the precise offline behavior would require an explicit logout rule or other direct evidence.