What is the ruling regarding someone who sees an ignorant person making mistakes in their prayer or fasting but does not object to them or teach them, even though they disapprove of it in their heart?

If you see someone doing something that invalidates their prayer or fasting, merely disapproving in your heart is not sufficient. Rather, it is obligatory for you to teach them if you are knowledgeable of the legal ruling. Sheikh Ibn Baz mentioned in Majmu' al-Fatawa that it is obligatory to try to correct mistakes in acts of worship through advice and by clarifying the legal ruling.