Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
The reason why the verse of Mubahalah challenge, no 61 of Surah Aali-Imran was revealed, according to the books of Tafseer (interpretation of the Qur'an), is that the Christians of Najran came to the Prophet (Sallallahu Alaihi wa Sallam) and started to argue with him about Isa (Jesus) and this verse was not abrogated.
The evidence is that many companions called to the Mubahalah (to pray and invoke sincerely the curse of Allah upon those who lie). Among them is Abdullah Ibn Mas'ood, (may Allah be pleased with him), who called to prove that the waiting period (Iddah) of a pregnant woman ends when she gives birth to the baby, not three months as is normal. Among them also is Ibn Abbas, (may Allah be pleased with him), who called to Mubahalah in the issue related to the reduction of inheritance's shares (when it is not possible for every heir to take his original share).
As regards whether this is permissible between Muslims, the answer is that it is permissible; the evidence is that the companions did so, as we already mentioned.
However, it is more appropriate for a Muslim not to invoke the curse of Allah upon a Muslim except after establishing the evidence against him, trying to eradicate the misconception, advising and warning him.
Allah knows best.