Is it obligatory for the congregant to correct the Imam if he makes a mistake in reciting Surat Al-Fatihah by reciting "الرَّحْمنِ الرَّحِيمِ" (Ar-Rahmanir-Rahim) with a Dammah on the letter “nun” in "Ar-Rahman"?

This is the hidden melodic error that does not invalidate the prayer, and it is not obligatory to correct the Imam therein.

Ibn Baz said: "The melodic error that does not change the meaning, such as reading 'Rabb al-ʿalamin' (Lord of the worlds) or 'Ar-Rahman' (The Most Merciful) with a fatḥah or a ḍammah, does not invalidate the prayer."

Correcting the Imam in this situation is permissible and recommended, even if not obligatory.

Al-Uthaymeen said: "If the Imam makes a mistake in recitation that alters the meaning, then it is obligatory to correct him. If it does not alter the meaning, then it is preferable to correct him, but not obligatory."