Can tourists study in public libraries?

I'll be in Tokyo for a few days for family vacation but I'll also need to do some studying. Can tourists have access to public libraries? Any fees or do we need to get permits? Thanks a lot.

Consult the library's webpage. Typically, fees are only if you want to borrow books, etc. However, don't expect electric plugs or wifi.

The library spaces are used for research purpose only.
You can not use there for your private purpose.

http://lib-www.smt.city.sendai.jp/guide/faq.html#usage01

入館するのに手続きもお金もいりません。 どなたでもご自由にお入りください。

And yes, the same is true in Tokyo, it would be surprising if it weren't... See here in Chiyoda for example:

http://www.library.chiyoda.tokyo.jp/guidance/space/chiyoda/

利用条件 どなたでもご利用になれます。貸出券は不要です。

As tokyo friend says, I have seen signs in some libraries here saying that the desks, etc. are only to be used with the library's own materials, and not with books, study materials, etc. of your own that you've brought in.

I imagine it might vary from library to library though.

Typically you have both types of desks, some "free" and some to be used only when browsing materies. But in any case, yes, best inquire when you're there...

This is largest in Japan.
http://www.ndl.go.jp/en/service/index.html
Also University of Tokyo Library one of historic one.
http://www.lib.u-tokyo.ac.jp/index-e.html
http://www.lib.u-tokyo.ac.jp/sitemap-e.html

Others in Tokyo.
https://matome.naver.jp/search?q=%E6%9D%B1%E4%BA%AC%E3%80%80%E5%9B%B3%...
https://matome.naver.jp/search?q=%E6%9D%B1%E4%BA%AC%E3%80%80%E5%9B%B3%...
http://tokyo-toshokan.net/index_statistics.htm
Use Google translate.

Basically if No read any library's books you should go to "Starbucks".
But in Tokyo(over 10 million living) some libraries accepting for 自習(jishuu/just for private study) visitors.
http://www.tosyokan-navi.com/list_p_p/tokyo/ure_jisyu.html

Anyone is free to enter most public libraries in Tokyo, and pick up a book, magazine or newspaper from the public shelves.

You need to be a resident to have a library card issued, and you need the card to borrow books to take outside.

But for example, in Chiyoda Public Library of which Firas posted a link, non-card-holders can ask to use seats/desks too. Internet access and electricity is available at those desks. Video games are prohibited. Public libraries throughout Japan usually works that way. Scroll down to "Facilities of the Chiyoda Library" for some details in English:
http://www.library.chiyoda.tokyo.jp/english/

Other great options can be the museums. There are often research rooms where anyone can ask permission to look through rare books and study them at a desk free of charge.

Just do what the locals do. Study in Starbucks and McDonalds.