I've experienced this repeatedly over the past week on Chrome for iOS. It seems some kind of JavaScript is handling touch/click events in a strange way.
When tapping on the "Developer Log In" button, the LDAP log in form appears but then immediately disappears again. The same also happens for the search icon and the search bar, they almost impossible to keep open.
I can sometimes get it to work when long-pressing on the button and then slowly dragging away from the button. Given it affects multiple buttons in Phabricator (not just our custom login), I assume this is an upstream bug.