No. It won't log you out if you are already logged on on a different device.
When using webmail (on a browser) if the email account is not being used and the email page is closed it would normally log you out automatically.
If the password is saved on the browser and auto fills the username and password for you, it would not allow you to log on until you enter the "new" password.
If you have signed on via an email client/app, if you change the password the app would not be able to automatically log on until the "new" password has been set up on the email client/app.
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