The subject header says it all.
For me it’s the tick-box on the login screen that doesn’t work on my iphone. Get the feeling that if the form controls worked OK, I could stay logged in on my phone.
I am also having some trouble with on Chrome 29m/Win7.
The HTML looks correct. It may be to do with the CSS “float” that positions the checkbox to the right of the submit button? Don’t have much experience developing for iOs so just a hypothesis 
Ah, at first I interpreted it as that the function in itself didn’t work; that you’ll get logged out after closing the browser.
Is it just the checkbox in itself that is not responding to input?
Or does anyone have problems with the browser forgetting your session even though the checkbox is checked?
Is it just the checkbox in itself that is not responding to input?
Yes. Tapping it doesn’t change its state.