%   #!/bin/sh
%
%   export curdate=`date +%d%m%y`
%   export reject=`date +%d`

Many /bin/sh shells do *not* support the bash-extension of setting a
variable and exporting it at the same time.

The date arguments are not quoted properly.  In this case, it's probably
going to work, but it's safer to use quotes.

    curdate=` date '+%d%m%y' `
    export curdate

    reject=` date '+%d' `
    export reject

%   mount /log-stale
%   cd /log-stale
%   mkdir HOSTS
%   cd /log-stale/HOSTS

No error-checking :(

I'd probably start with something like:

    err()
    {
        errcode="$1"
        shift

        if test -z "${errcode}" ; then
            errcode=1
        fi
    }

and use it every time an operation fails:

    stalelogdir="/log-stale"
    hostdir="${stalelogdir}/HOSTS"

    mount "${stalelogdir}"
    if test $? -ne 0 ; then
        err 1 "${stalelogdir}: cannot mount"
    fi

    if test ! -d "${hostdir}" ; then
        mkdir -p "${hostdir}"
        if test $? -ne 0 ; then
            umount "${stalelogdir}"
            err 1 "${hostdir}: cannot create directory"
        fi
    fi

%   for i in ls /log-fresh/HOSTS/*/*/*/ ; do if [[ ls $i != /log-fresh/HOSTS/*/*/reject ]]; then mv $i /log-stale/HOSTS/ fi ; done

Indentation is 90% of the readability of a source file.  Please wrap
lines to a more reasonable length.

Another tricky point here is that if there are too many matches in the
filename wildcard, you will hit command-line limits.  It's usually
better to feed filenames one by one to a reader loop:

    find /log-fresh/HOSTS -type d | \
    while read dname ; do
        # Perform something in directory "${dname}"
    done

%   tar -cf logs_$curdate.tar /log-stale/HOSTS/
%   bzip2 logs_$curdate.tar
%   rm -rf /log-stale/HOSTS/
%   cd /
%   umount /log-stale
%   exit 0

No error checking here either :(