rsync: skip files for which I don't have permissions

Rsync doesn't have an option for this. I see two solutions. One is to parse rsync error messages; this isn't very robust. The other is to generate a list of unreadable files to filter.

cd /source/directory
exclude_file=$(mktemp)
find . ! -readable -o -type d ! -executable |
  sed -e 's:^\./:/:' -e 's:[?*\\[]:\\1:g' >>"$exclude_file"
rsync -rlptD --exclude-from="$exclude_file" . /target/directory
rm "$exclude_file"

If your find doesn't have -readable and -executable, replace them by the appropriate -perm directive.

This assumes that there are no unreadable files whose name contains a newline. If you need to cope with those, you'll need to produce a null-delimited file list like this, and pass the -0 option to rsync:

find . \( ! -readable -o -type d ! -executable \) -print0 |
  perl -0000 -pe 's:\A\./:/:' -e 's:[?*\\[]:$1:g' >>"$exclude_file"

I made a simple workaround for this specific situation:

rsync --args || $(case "$?" in 0|23) exit 0 ;; *) exit $?; esac)

This returns 0 if the returned code was 0 or 23, and returns the exit code in all other cases.

It is important to note, however, that this would ignore all Partial transfer due to error errors, not just permission ones, since it will catch everything that exits code 23. For more information about rsync status codes please refer to this link.