What will happen when a user try to login after his home directory get full?

It depends on how the user login, with (telnet,ssh, rlogin) or login through desktop session.
For telnet login, user will be able to login but he won't be able to create any file.

For desktop the answer could be yes and no. Some desktops like GNOME could point the creation of the files to /tmp which would still allow the user temporary files to be created and have a successful login. These would generally be the unix socket files for IPC between applications and the underlying framework. What could get blocked is the further use of the desktop if any configuration changes are done as they would need to be stored in the $HOME.

In earlier versions of GNOME, $HOME was used in which case the login would fail but later they moved on to use $TMPDIR (generally pointing to /tmp or /var/tmp).