Back when I was using Battle.net to sign in, I would always stay signed in at bungie.net, but ever since I'm logging in with Steam, it never remembers me.
I'm using the latest production version of Firefox on Windows 10. The browser is NOT set up to clear cookies upon closing and remembers my logins on all other sites. I'm always logged in on the Steam website. Also, this doesn't just happen when I close the browser. If I close bungie.net after sending this message and come back in an hour or so, I will have to sign in again, even though the browser will stay open.
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I imagine the issue is with Steam expiring sessions after 24 hours. I'll occasionally be able to log back in without getting a code from the app, but that's very rare.