Furuhon o kattara, page no aida ni 1 man yen satsu ga hasamatte ita.
Why there is "ita" in this sentence? As i know, Imasu is for non living thing.
About your question, the bill is stuck in the pages.
So the word that refers to the action of being stuck is "hasamatte ita." The translation more or less would be "had been stuck."
"Hasamatte ita" is the conjugated form of "hasamaru" which means "to get stuck." But the bill was already stuck, making it "hasamatte iru." However, the sentence refers to something in the past--therefore "hasamatte ita."
"Ita" means "to be," as in "the state of being"-- so it means that the bill had been in the state of being stuck in the pages.
Now, I don't know exactly what you mean by non-living things, but "imasu" is the polite version of "iru." The past tense of "iru" is "ita" and the polite form of that would be "imashita."
Hope this helps!
I agree with the first comment about the importance of learning the actual writing system. Learners of Japanese who use romaji are at a big disadvantage. Fewer materials, less chance to communicate with native speakers, and particularly unpleasant outcomes when it comes to actually making progress in all aspects of the language. In Japanese your pronunciation will suffer very much if you learn by reading romaji.
but to answer your question. It's because the bill is in the state of being stuck between the pages of the book. V-te + iru is a grammatical construction. In this case iru doesn't refer to whether the money is living or non living. It is just part of the grammatical construction. Here is a site I just found with examples written in romaji as well.
There is a lot of confusing grammar in Japanese where the same or similar words appear but the meaning or purpose of the word isn't the same. [emoji]