What are the teachings of Hinduism for dealing with other religions and their followers? To be more precise, what does Hinduism teach its followers about how to treat other people like Muslims and Christians?
Hindu Dharma is non-exclusive. I.e it does not claim it possesses the exclusive right to grant salvation/liberation. It recognizes that every path can lead to the truth.
"ekam sat viprāḥ bahudhā vadanti"
-Rig Veda (Book 1, Hymn 164, Verse 46)
This means "The Truth is one; the sages talk of it in many ways."
Hence Hindus should treat others with "Mutual Respect". Note the word "mutual". Hindus should respect other's belief and their paths and also expect others to do the same to them. But this might not be possible with religions which do not offer mutual respect.
There is no concepts of False Gods and false prophets in Hinduism. It is more of a self-exploration of the mind(adhyatmik) than an organized religion. Hence the individual's experience(anubhava) is given respect, which again emphasizes respect to other religions. Though that does not prevent Hindus from engaging in philosophical debates with others. Adi Sankara is said to have vigorously debated Mimaamsa-vaadis & Buddhists. The debate is done with respect and not with contempt.
The teachings of Ramakrishna And his disciple Swami Vivekananda , have explained how Hindus should treat people of other religions.
According to Hindu belief ,God is one but there are many ways of reaching Him .Just as the Ocean is one ,but there are many rivers falling into it from many different countries and many different directions.Ramakrishna himself practiced many faiths -- including Islam and Christianity -- and finally discovered that they lead to the same experience.
Theoretically Hinduism does not make any distinction between different faiths -- treating them like different rivers trying to reach the Ocean.
In practice , Hindus do make a distinction -- between rivers (faiths) that will one day end up in the Ocean (God) and rivers (faiths) which might end up in a Desert and dry out (Suicidal paths) .Use of Force and Violence is Justified in dealing with people following Anti-social and Suicidal paths --to make them Socially useful.This is described in the " Bhagavat Gita".
Hinduism is not as much of a "Religion" as a conglomerate of different philosophies mixed in. These philosophies include Samkhya shastra, Advaita vedanta, Dwaita vendanta, etc. and even some nastika schools like Buddhist and Jain philosophies.
Hindu in fact is used by foreigners to describe the people who stayed across the
Sindhu river. So yes, throughout history, Hindu religion has been the one to mutually interact with other lots of other religions and cultures including the Greeks and Persians. Because of its outstanding religious tolerance and its own integrated (though diverse) social structure like the varana-ashram system, Hinduism has survived the onslaught of myriad of foreign invasions throughout its History.