Is it permissible to recite the morning and evening adhkar while performing other tasks such as housework or driving?

There is nothing to prevent a Muslim from reciting dhikr (remembrance of Allah) while driving, because the Prophet, peace and blessings be upon him, used to remember Allah in all his states. However, what is important is for theaker to be present in heart. It is better for a Muslim, if they wish to recite dhikr, to dedicate time specifically for it.