If I want to transfer a large file between two computers, which are connected via LAN, would going through the network (setup as gigabit ethernet) be faster than using a direct Firewire 800 connection between the computers? If Firewire 800 is much faster, can I setup a LAN with multiple computers with Firewire 800?
In general, GigE should be faster than Firewire 800. There are some variables at work that may make FireWire faster, but except for Ethernet congestion from other traffic they're edge cases. If you're looking to connect more than 2 hosts, Ethernet is by far the better choice.
Gigabit is in much faster unless you provide it via a slow expansion method such as PCI. Firewire is only 786 Mbits, but if you have significant traffic on your network you could conceivably get higher speeds from firewire.
In general you should do whatever is easier for you, since the raw connection speed of both technologies is fairly similar.
If you already have a gigabit network set up, don't worry about creating a new firewire connection. If you don't, maybe (Macs for example) Firewire is easier. But all that depends on your environment and the kind of software/hardware you use.
More specifically, the absolute transfer time for your file depends not only on Firewire vs. Ethernet, but also on the protocols. With Ethernet you'd usually use IP. But with TCP or UDP? TCP is good for unstable connections, while UDP has less of an overhead. Even if you decide on TCP, Windows file sharing, just as an example, works rather well if you want to transfer lots of small files, whereas FTP is pretty good at transferring fewer but larger files.
Other replies in this thread make excellent points. I would add this one to the mix: make sure that the infrastructure that supports your gigabit ethernet network is capable of jumbo frames and that jumbo frames are enabled everywhere. That includes the ethetrnet cards on all relevant computers, servers, and ethernet switches.
I learned about this the hard way when I decided to discontinue using my firewire 800 drives in favor of a NAS connected over Gigabit ethernet to my 27" late 2009 iMac. The NAS supports jumbo frames, the switch supports jumbo frames. Unfortunately, the ethernet card built into the late 2009 iMac does not. :-( It's fast, but it's nowhere near as fast as firewire 800 drives connected to the Mac.