Kdjykuj

Can I visit the Trinity College (Cambridge) library and see some of their rare books: Newton Principia, Ramanujan, etc., as displayed in the "The Man who Knew Infinity" movie?

The Wren Library of Trinity college is open to the public from noon to 2pm each weekday throughout the year and also from 10:30am to 12:30pm on Saturdays during university "full term". It's closed at Christmas and Easter.

Their website says, "There are six exhibition cases in which a small fraction of the Library’s treasures are on display" but it doesn't give more precise information. You could ask the Trinity College sub-librarian what is currently on display; their email address is included on that page.

I expect that Newton's copy of Principia would normally be on display because it's so famously associated with the college. However, it's currently (until August 2019) on display in a public exhibition at the University Library, which is open 9am–6pm Monday–Friday and 9am–4:30pm on Saturdays.

The Wren Library's Ramanujan collection has been digitized and is available online. Of course, that's not the same thing as seeing the original documents. Newton's Principia and Opticks are also online.

NB according to comments on this answer, what I've written here is only accurate for the main university library and does not apply to individual college libraries. Please refer to this answer for the correct information for Trinity College.

The library website states (emphasis mine)

All members of the University of Cambridge are welcome to use the Library. Please note that you may need to bring some supporting documentation. See ‘Eligibility’ for full details.

Academic staff, research students and taught postgraduate students from other universities can apply to use the Library for reference.

Undergraduate students from other UK universities can apply to use the Library for reference outside University of Cambridge full terms.

Private and business researchers may apply to use the Library for reference if their research requires access to materials held at the University Library.

So unless you fall under one of the first three categories, its unlikely that you'll be able to visit "just for fun", you'll probably need to justify why you need access to the library. There are contact details on that page so you can get in touch with the library and confirm whether you'll be allowed access.

The good news is that, according to this, if you are allowed access as a private researcher, you'll get access to the Special Collections, which is where such rare books would be likely to be found. The searchable library index should be able to confirm this.

Note that, as you're probably aware, then in the film, the protagonist is an academic at the University and so would probably be allowed greater access than the general public.