Well... I made a web page to demonstrate the problem for you, but both worked fine (i.e. with and without the smart quotes).
When I first began working on the Web in the late 90's I composed most of my posts using Word, then copy and pasted to my site. That was when I first noticed the problem. I used to I got a little box where the quotes were supposed to be, and so I always kept "smart quotes" turned off. I just assumed the same would be true with Open Office. We are having a storm right now so I decided to use my lap top for a study I am working on instead of my desk top in case we lose power, and it uses Open Office. Apparently this is no longer a problem.
Thanks for your help.
Many if not most web browsers interpret the smart quotes (the curly " marks that auto-replace the normal quote marks) with broken icons.
Can you give an example of this? I've not noticed it before.
Also, I'm not entirely clear on what the quotes in Writer have to do with your web browser. Can you explain a little more about what you're trying to do?