How I can check what takes disk space in /tmp?

The first command indicates that /tmp is actually on the same filesystem as / (ie, everything else). If your root partition is full, it could be that other stuff (such as /var/log) is taking up space.

A decent way of finding things is to do

du -sc * .[^.]* | sort -n

to find what directories are big. Then you can continue to cd into lower directories and rerun the command to narrow things down.

On a recent distro :

du -sh /tmp/* | sort -h

On a older distro :

du -csm /tmp/* | sort -n

Check if you didnt run out of inodes.

# df -i
Filesystem            Inodes   IUsed   IFree IUse% Mounted on
/dev/sda3            1236992 1236992       0  100% /
tmpfs                1007716       1 1007715    1% /dev/shm
/dev/sda1              38456      45   38411    1% /boot