Why do Hindus have no restrictions on going to churches or any other religious places?
Because Hinduism is not structured like Islam or Christianity. There is no 'one' supreme god or 'one' single book to guide you. Everyone is free to find their own way to the ultimate freedom (moksha) and most of the religious books (Vedas, Upanishads, Geeta etc.) prohibit harming other living things. As long as you don't hurt anyone, you can live even without praying and not be condemned. If however you wish to pursue spiritual path to achieve moksha, you are free to worship any god/force/inspiration in a harmless way.
I am an atheist and have learnt a little about a few religions. I have come to believe that Hinduism is the next best thing to atheism and agnosticism.
I had answered this question 6 years ago, when I was way more ignorant (than now :) ), and largely unaware of my ignorance, and had many presumptions, and misconceptions. (Reading this extremely narcissistic, and idiotic line makes me cringe every time: "I am an atheist and have learnt a little about a few religions".. )
Now (year 2020), I would't even dare to attempt answering this question, as I wouldn't know what you mean by Hinduism, atheism, agnosticism etc. , let alone compare it to other religions (which I know even less about), because these concepts are defined / understood differently by almost everyone. Specially defining hinduism is even more difficult cause there isn't one definition / practice / book / institution accepted by ALL.
Also, though generally Hindu population (or a large section of it, if not the religion itself) do not have restriction on visiting other religious places, there are few Hindu temples, which do not allow non-hundu people to enter. So its not as open a culture / religion EVERYWHERE in India as well.
I did not want to delete / edit original text in my answer because it is an accepted answer, and has been upvoted a few times, so changing it would feel like cheating, even if the answer turns out to be inaccurate, (or as some comments below argue, completely wrong).
My original answer, would probably, still have some relevance, if you consider the question only asks why "Hindus" (not hinduism) have no restrictions, and the fact that most Hindus (even some religiously sincere ones), haven't read, are aware of, or even considered the question of what should be, THE authoritative book / god / person / practice etc. to follow.
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