The replies above are useless for this very common complaint. Here's how to solve the problem:
PCs: Copy the highlighted text. Paste it with the icon that has the arrow on it. This preserves formatting EXCEPT highlighting is removed.
Macs: Copy the highlighted text. Use "Paste Preserving Formatting". Same result as above.
MICROSOFT: Get on the stick and address this pernicious bug! Remove Highlighting should be made to work.
In Word, there are several ways to accomplish the same thing.
Just to be clear: neither highlighting nor shading is buggy in Word (up to and including Word 2010, at least—I haven't tested Word 2013 yet). The application is working as designed.