There is an addin called Xmarks from Marvasol Inc. which might do what you want, depending on your definition of "persistent session" (see below). It can sync browser tabs between instances, either manually or automatically on Firefox startup.
The content of the tabs won't be exactly in sync because the HTTP requests are submitted separately from each computer, so if you are expecting the content itself to be requested only from one computer and shared between the computers, this extension does not do that (and I am not aware of any extension that would, short of doing something like VNC/RDP/NX).
I am also not aware of a built-in way to do this in Firefox without addins.
By "persistent sessions", what do you mean? There are several things that can be implied by this terminology. So here is some additional information if you want to dig into the details of exactly what you are asking for in an update to your question, so that we can more clearly address your needs:
- Do you want to copy the session cookies / persistent cookies from one machine to the other automatically? (this is feasible, but I don't believe that Xmarks does this. You'd also have to migrate over any HTML5 data, Flash data, etc. if you want specific functionality of advanced applications to remain consistent between computers.)
- Do you want to maintain the same HTTP connection to the server from one endpoint and maintain that "live" session from one box to the other? (kind of like teleporting a VM, but involves one box obtaining the IP address of the other and some other very complicated dynamics)
- Do you just want to store the URLs of the tabs themselves and have the URLs be re-requested on the remote host? (this is what Xmarks does)