I've recently discovered a Ginger Turmeric Herbal Tea that I love. Is there any problem with drinking 5 cups of this stuff a day?

Go for it! Ginger and Turmeric are two of the most healthful herbs once can consume and there is no danger from consuming large amounts, so have as much as you desire! See http://www.whfoods.com/genpage.php?tname=foodspice&dbid=72 and http://www.whfoods.com/genpage.php?tname=foodspice&dbid=78.