There is no way to actually make IE not show the open/download prompt when clicking a linked file, as that is a security measure designed to stop sites from downloading unwanted content on your machine. Just as you can, any user could alter their settings to always open FLV files, but making them do it should not even be considered. So if you want to serve video files to your users, you'd want to use an actual video player, not just a link to the file.
Html5 video tag offer plenty of options and is supported by most modern browsers, especially if you use the WebM format for your video. Here is a nice guide on how to make your own design, as all browsers display the html player differently.
The best solution, though, would be to use a player that uses both html5 and flash, so it can support all browsers, including mobile platforms. iOS, we all know, refuses to play flash except through special browsers that download the content you are viewing to their servers and send it to you in a different form. Something like FlowPlayer or MediaElements would be perfect.
The below part is not a valid answer to the OP's question, as he wants to serve FLV video on visitors. It is, however, still usefull to people who wants to configure their own IE browser to always open FLV files by default.
Save the FLV file instead of opening it, then click "View Downloads". Right click on the file and uncheck "Always ask before opening this type of file". From now on, all FLV files should open by default.