I've done lots of searching but I haven't found anything like it. Is there a chest protection mod similar to LWC for Forge?

For non bukkit I don't believe so but its not hard to make one built into chests

Industrial Craft 2 has a 'Personal Chest' that only the owner of it can access.  It is quite simple to make.

 

The problem with having unbreakable, inaccessible by everyone else chests though is that people start to make unbreakable walls with them...

Forgot about that little fact, got rid of thw notion on my server. People tend not to make unbreakabe walls when admins say no with more than words. Though some form of protection is needed inorder for player to have full protection. Say they can be broke just don't drop anything or you need special tools to move them.

That requires administration, better to have limitations in the game itself. 

 

I personally think that the Factions Bukkit mod does it very well for servers like that, if you want to access a chest then you need to invade their defenses and get to the chest. 

Factions loved that plugin til my admin couldn't interact with faction areas

 

Yes limit in a protection mods are needed but at the same time actual protect. For example no need for a lock if it break with no work.something I was think of if I was to make a mod like this is lock picks. With tiers of protect all the way up to admin protection.