1
general::
Refuse to exert oneself to help or perform an action. For example, Dad won't lift a finger to help them financially, or Early in the war, America officially would not lift a finger. [Mid-1900s]
American Heritage Idioms
2
general::
Phrase(s): not lift a finger (to help someone) [and] not lift a hand (to help someone)
Fig. to do nothing to help someone. (The someone is anyone with the negative.) • They wouldn’t lift a finger to help us. • Can you imagine that they wouldn’t lift a finger? • Sally refused to lift a hand to help her own sister.
McGrawhill's American Idioms And Phrasal Verbs