Aircraft do have LAN Networks for their sensors to minimize wiring (and consequently weight), however I would assume these are closed circuit networks and NOT connected to the IFE system.
In terms of the FMS system on aircraft according to this they do have databases however I dug into it a bit and it seems that based on this walk through on updating you need to physically connect to the device to change that database (for Rockwell Collins Equipment). It also seems that the Honeywell FMS computers have a similar way of updating via USB. While I cant say for certain it does not seem the FMS is on any kind of network aside from its own.
The big question is autopilot/FBW systems and can they be overridden or tampered with. It seems that Boeing-Honeywell has a patent on some remote flying systems but has not yet implemented it on any aircraft. I am still looking for some solid evidence that the systems have a point of access on the network. It is clear that they can be altered if you can physically connect to them for update purposes. Keep in mind you would have to load them with false data.
This seems to be the autopilot for the 777 however Rockwell Collins does not seem to allow you to get the info on it very readily. Maybe someone else on here can comment on how its set up.
This nice paper overview of FBW systems on a 777, in section 11.4.4 lists the connections the FBW system has to other systems on the plane. While this is by no means an official document (at least I don't think so) it does not list the IFE system as a connection and is pretty clear on the fact that everything connects over the ARINC 629 style data busses.
It does appear that ethernet technology is making its way onto aircraft however it seems that the tech still uses ARINC Specification 664. However page 2 of this spec documents seems to imply that through a "gateway" the system is connected to the "internet". However Page 38 clearly states
Although AFDX/ARINC 664 is a closed network, UDP port numbers
should be selected from the Dynamic/Private range of numbers. The reason
for this is that there could be potential conflicts with the standard port
number assignments when a gateway is used to communicate between the
AFDX network and the Internet
So it is connected to the internet but the flight control systems are on a closed network. They also advise you to use the private range of ports to avoid any possible conflicts. Here is another good paper on the technology. It does seem that they are moving towards some internet like devices on aircraft but it also seems that they will remain closed networks. Companies are looking to consume some modern tech for an age old application as its simpler and easier to use existing high speed bus technology than reinvent the wheel and make a new custom bus. Just because something has a router and ethernet cables does not mean its connected to the internet!