What is the long-term capital gains tax for South Korean companies
when they sell shares they own in other companies?  In other words, if
a Korean company sells some stock of another Korean company, what is
the effective tax rate it must pay on any gains?

It will be 20% or 30% depending on the duration held.

In case of unlisted shares, if a major stockholder of a large
corporation transfers them after retaining less than a year, a tax
rate of 30 percent shall be applied. If such stocks are retained for
more than a year, a tax rate of 20 percent is applied. Minority
stockholders of large corporations transferring unlist-ed stocks shall
be liable to a 20 percent rate, regardless of the period retained. If
a small and medium corporation transfers its unlisted stocks, they
will be liable to 20 percent tax, regardless of its period retained.

Listed stockholders, in principle, are normally not subject to tax.
If, however, a majority shareholder of a large business transfers the
shares after holding on to them for less than a year, he or she will
be liable to a 30 percent rater of tax. If he or she retains the
stocks for more than a year, the tax will be 20 percent.


Please use this link for your doubts:
http://kilc.kita.org/kilc/english/taxsystem/taxsystem.jsp#1

Hope this helps!