I took a look at the app store and I found a few apps that can sync with Firefox and some other browsers. Below are some details.
i-Pusher ($2.99, no ratings): Submit various content from your favorite desktop browser into your i-Device in one click...you will instantly receive a link, text, image or youtube video link on your mobile device. Personal Note: You do have to install the appropriate extension in your desktop browser.
Firefox Home (Free, 3/5 stars) Firefox Home Provides access to your desktop history, bookmarks and open tabs on your iPhone. Personal Note: Not a web browser itself. I think this is close to home of what you need. I played with this before and I was able to continue browsing my opened tabs on my desktop Firefox using its Firefox Sync feature. http://itunes.apple.com/us/app/firefox-home/id380366933?mt=8
360 Web Browser | Download Manager and Firefox Sync (99¢, 4/5 stars) Personal Note: This is actually a third party browser with many desktop like functionality like add ons and download managers with many configuration settings. It uses Firefox Sync, found in Settings in the app. I just reread its description and apparently it can also do Flash Videos using a CC Plugin. http://itunes.apple.com/us/app/360-web-browser-download-manager/id387737097?mt=8