Even if Stay signed in is checked on the login screen by default, Yahoo still might prompt you to log in each time you want to check your Yahoo Mail. If this is the case, then your browser isn't saving login cookies, which are bits of data that let Yahoo know you're a return visitor. You can stay logged in to your Yahoo Mail account by making some adjustment to your browser's security settings.
Information in this article applies broadly to using Yahoo Mail on any device with any browser.

When You Have to Log In to Yahoo Mail

The cookie your browser saves when you visit Yahoo Mail applies only to the browser and device you were using at the time of your visit. As long as you visit the login page with the same device and browser, you shouldn't have to log back in. However, if you attempt to log in with a different device or browser, Yahoo won't find the login cookie, so you'll have to enter your username and password.
If you are using the same device and browser, but you are still prompted to log in, then something or someone deleted the cookie in your browser that would log you in automatically.

How to Stay Logged In to Yahoo Mail

You can prevent your computer from deleting your browser cookies, including the one for your Yahoo Mail login credentials, in a few ways.

Select Stay Singed In

When you log in to Yahoo Mail, confirm that the box beside Stay signed in under your login credentials is checked.

Don't Sign Out

Don't click Sign out in the box that appears when you select your name at the top of any Yahoo page.

Don't Delete Cookies

Do not clear your browser cookies manually. Also, check your browser's settings to make sure you don't have it set to delete cookies when the browser window closes. If you run browser extensions and anti-spyware that automatically clears your browser history, consider disabling them, or make an exception for the yahoo.com domain.

Don't Use Private Browsing

For enhanced internet privacy, you can use your browser's private browsing feature to visit websites without storing cookies on your computer. That way, your browser won't track your internet history, but you'll have to sign in to Yahoo Mail each time you visit. Using this feature frequently may explain why your login information isn't saved.
The various browsers have different names for the private browsing feature:
Air

Can't Sign Out Of Yahoo Mail App

Is miracle box legit. If you'd prefer not to log in to Yahoo Mail each time you visit, don't use private browsing.
I had a similar issue. The Gateway Login Screen just kept poping up. Remote desktop the logon attempt failed rd gateway.
From: Yahoo! Account Services <*** Email address is removed for privacy ***>
To: Ken Sent: Sun, April 18, 2010 10:35:06 AM
Subject: Re: A Message From Yahoo! Customer Care- 10347904 (KMM165315036V538L0KM)

Hello Ken,
Thank you for writing to Yahoo! Account Services.
I understand you are still being looped to the enter password screen
even after entering your ID and password. I apologize for any
inconvenience this may have caused you.
We were unable to reproduce the problem you described. Each time we
tried signing in, we were able to access your account on the first try.
We tried signing in on both Yahoo! Mail at:
http://mail.yahoo.com
and on My Yahoo! at:
http://my.yahoo.com
The problem may be due to the configuration of your computer.
Please note that a difficulty setting your user cookie may not
necessarily affect other Yahoo! accounts that you may be accessing from
the same computer.
You may want to try signing in using a different web browser (Internet
Explorer, Firefox, Chrome) or computer. We recommend that you contact
your browser's equipment manufacturers or Internet Service Provider help
desks for information regarding this problem.
Also have temporally deactivated your account to clear the login issue.
You may have received an email notice of this.
In the meantime try the same steps at a later time, on another computer
or try using another Yahoo! supported browser. You can find our most
popular FREE software, now all in one place by visiting:
http://downloads.yahoo.com/
I will report this into our engineering team in which they will begin to
investigate. At a later date it may be necessary for Yahoo! to contact
you proactively and collect more information. If that step is necessary,
you'll see a reply to this email asking for more specific info. We
appreciate any assistance you could provide.
Again, I want to thank you for bringing this to our attention and want
to apologize for any inconvenience this known issue may have caused you.
We would like to obtain more information from you regarding this problem
and we would like you to call us at your earliest convenience
at 503.615.390.
We hope this is resolved for you soon, so that you may continue to use
your Yahoo! account. Please contact us if, after speaking with the above
individuals, there is something we can do to assist you in fixing this
problem.
Thank you again for contacting Yahoo! Account Services.
Regards,
Dalton
Yahoo! Account Services Customer Care