Is it possible to have multiple machines running Firefox (possibly connected by sync) that can resume certain stored sessions of browsing tabs, that can be created/update on one machine, then simply replicated onto another machine (either while the first session is active, or if it has been deactivated and the session data stored on the server)?
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: