I have 2 pages of html, one is test01.html and another is testing2.html files.
My question is: In test01.html page I have some text fields,
That's something you can do with Ajax, but it's a bit heavy duty unless you know how that works.
or can we do like this, I want to hide a button in the test01.html on clicking on the button in the other page
I don't understand; and on reading your first post again, I'm not sure even Ajax could do that. It will pull content from another page into the current page, but not the reverse.
If page 1 uses scripting to open page 2, and they're both on the same domain, then the second page should be able to communicate with the first by using parent.
See this Cross-window scripting page for details.