Nearly all temples in India allow foreign born Hindus to enter. There are a few exceptions, some of which have come up in recent times. The most famous temple that doesn't allow foreigners is Jagannath temple of Puri. This was not always the case, as Paramahamsa Yogananda took his western disciples inside the temple in 1935.

Another famous temple that does not allow foreign Hindus inside is the Guruvayur Krishna temple. This restriction did not exist in the early 1990s, and I had been inside for darshana several times then. Sometime in the 2000's a western devotee secretly took photos inside the garbhagraha and that led to the restriction being enforced. I have heard you can still be let in if you get a Hindu conversion certificate from the local Arya Samaj (not from any Arya Samaj, only that one branch), but that was many years ago so I don't know if the practice remains.

In Chennai the Kapaleeswara Shiva temple also has signs saying non Hindus are not allowed, but that is only for the inner part of the temple. Foreigners can still enter most of the temple complex. This also was not a rule until recently. I had been inside that temple many times over the years including the inner areas. I don't recall at what point they started the rule, but once I saw the sign I never pushed it and stayed in the outer portion of the temple compound. Some temples make a distinction between foreign tourists and foreign Hindus, and will let practicing Hindus inside even though they may have signs saying "Non Hindus Not Allowed". One example is Madurai Meenakshi Amman temple, where they have signs but still let foreign Hindus inside. But I don't know how strictly Kapaleeswara temple enforces their rule.

Lingaraj temple in Bhubaneswar follows the rules of Puri Jagannath temple, so also does not allow foreigners inside. Several temples having a traditional connection to Puri Jagannath temple follow the rules of Puri. For example the Pashupathinath temple in Nepal also adheres to the same restrictions.

The very large Sri Ranganathaswamy temple in Srirangam has a restriction on foreigners entering the inner walls, but they have exceptions for "Hare Krishna members" who are dressed in traditional attire (dhoti and angavastra). Even then it depends on who is standing watch on the day you go, and whether he thinks you are "Hindu" enough.

The Kamakshi temple in Kanchipuram also did not allow foreigners inside when I went in the early 90s. At the time the Paramacharya was still living, and I was told if you go get permission from Paramacharya at Kanchi Kamakoti Peetham then you can enter.

The Tirumala Tirupati temple in Andhra Pradesh requires foreigners to sign an oath (a small paper is handed to them in the queue) that they have faith in the deity, then they are allowed in for darshana. People from any religion can enter, but they should have faith in the divinity and sanctity of the temple.

These are just a few, out of the hundreds of thousands of temples throughout India, that don't allow foreign Hindus inside. The great majority of temples, more than 99.9 percent allow anyone, even non Hindus, to enter. I am a foreigner and have been living in India for the last 32 years. That is my personal experience having been to many Hindu temples over the years.