What is the ruling on charity that was collected for a sick teacher, but she refused it, and then it was given to the donor's poor sister?

What your relative collected for her colleague is either a voluntary charity (Sadaqah (definition) Nafal) or Zakat (definition). If it is a voluntary charity, she must inform the donor that the recipient did not accept the charity. The donor then has the option to take it back or give it as charity to someone else, which is preferable. If it is Zakat, it is better to consult its owner to consider its disbursement. So, if it was a voluntary charity, and the donor approved of its disposal, it is valid. Otherwise, she is liable for what she gave in charity. However, if it was Zakat, and her sister was eligible to receive Zakat, then she is not liable for anything.