Cat, Grep, Redirect Output.... Blank File?

This happened because the first thing > does is to create the file it wants to write to - and if the file already exists, its contents will be deleted. (Also, there's no need at all to use cat in your statement since grep works on files, not just on STDIN.)

The correct way to do this is to use a temporary file either to read from or to write to. So either

cp /opt/webapplications/Word/readme.log /tmp/readme.log
grep -v 'Apple' /tmp/readme.log > /opt/webapplications/Word/readme.log

or

grep -v 'Apple' /opt/webapplications/Word/readme.log > /tmp/readme.log
mv /tmp/readme.log /opt/webapplications/Word/readme.log

would work.