Applying a bookmark shouldn't cause the heading to change at all. (Except in Design view it looks like a hyperlink. )
If you put a bookmark on say some Normal text, does it happen too?
Also, maybe post your stylesheet - at least any entries containing H3 - maybe something there is affecting it.
This happens in standard compliance browsers, because the syntax is incorrect. If you specify an anchor, You must not include text bewteen the <a> tags. The correct syntax is:
You apply this in RH by putting the cursor on a random spot (do NOT select any text or image) and click INSERT | BOOKMARK. You bookmarks will then appear correct.
Note: Firefox is case sensitive, so make sure your links are correct.