MAJOR UPDATE (July 17, 2008)
According to the Firefox 3.0.1 release notes this bug has been fixed.

:::Original Post:::
I spent a lot of time looking for an answer to this problem, with no good answer other than to completely uninstall the network-manager package. Unfortunately, this wasn't really an option for me. I am on dial-up, but sometimes use the wireless at the library or at someone's house. So, network manager had to stay, but it was extremely annoying that every time I opened Firefox, I had to click File > (uncheck) Work Offline. Well, here's a solution that worked for me that allowed me to keep Network Manager installed, and make Firefox understand that even dial-up, slow as it might be, is still an internet connection.

If you open /etc/dbus-1/system.d/NetworkManager.conf as root (sudo nano /etc/dbus-1/system.d/NetworkManager.conf) and change a couple things, it might work for you.

There are three (3) places that you will find this:
<allow send_interface="org.freedesktop.NetworkManager"/>
These are found under:

<br />
<policy user="root">
  .......</p>
<policy user="haldaemon">
  .......</p>
<policy at_console="true">
  .......

What to do here is change "allow" to "deny" for all three entries. (a restart may be required for the configuration file to be reloaded.)

Apparently, the "allow send_interface..." ends up telling Firefox that there is no network connection for whatever reason. From what I have seen, this can also solve problems in other programs in K/Ubuntu 8.04 such as Evolution. (I wouldn't know personally, because I am still using 7.10 with a manually upgraded Firefox 3.0 and don't even use Evolution)

Until this, I figured the options were to wait for either networkmanager 0.7 (currently 0.6.5) or Firefox 3.0.1 to be released with the bugfix (hopefully). I'm not that patient though...

This SEEMS to be a harmless trick in the configuration file with no ill effects on the system, but if network connectivity is changed in a bad way, you can try changing the entries back one at a time to see where the problem is and/or which one specifically helps Firefox while not effecting anything else. Please post any failures and harmful effects here.

UPDATE:
There is indeed an ill effect. In my case, knetworkmanager is no longer actually functional, such as if I want to connect to my home wireless network for files, then disable and connect to dialup, it will not allow it because it is still trying to connect through the wireless, which I turned off manually by using the switch. In other words, this fix pretty well disables networkmanager. If anybody knows of or finds a fix for this, please post it here.