Can a 16-year-old move out on her own if her mother says it's ok although she lives with her guardian?

No, if there is a court appointed guardian, that person has legal precedence over a parent. The guardian has the legal right to make decisions on the part of said minor w/o input from either biological parent. Even if the guardian agreed to the minor leaving the residence, it would be necessary to obtain the permission of the court that granted the guardianship.

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