So I finally decided to have the interact flag set to deny in my town and, given how fastidiously meticulous I am, I decided to test it thoroughly. So I placed every kind of door, trapdoor, gate, etc that could be interacted with and paid a couple people not in my town to test them. Here are the results.

For every block/entity that could be lwc protected, I placed two: the first one without any protection, the second one with the /cpublic protection. For every block/entity that couldn't be lwc protected, I placed only one (without protection, of course).
Here is the list of all the blocks/entities that I've had tested: in red only what could be interacted with.

• all 6 wooden doors, /cpublic and not
• iron door, /cpublic and not
  
• all 6 wooden gates, /cpublic and not
• wood trapdoor, /cpublic and not
• iron trapdoor, /cpublic and not
• cauldron
• cake
• wood button
• stone button (cpublic and not)
• wood/stone/gold/iron pressure plate
• lever, /cpublic and not
• tripwire
• boat
• cart
• crafting table
• chest, /cpublic and not
• trapchest, /cpublic and not
• enderchest, /cpublic and not
• furnace, /cpublic and not
• brewing stand, /cpublic and not
• trade sign, /cpublic and not
• all ExtCreations signs, /cpublic and not

Here's my comment.
Boat and cart make totally sense: those are not even blocks.
The crafting table and all ExtCreations signs make sense too: they are blocks, but they cannot change state, so why forbid them?
Trade signs are a bit unusual: they do change their content when interacted with, so one would expect them not to work with nonmembers, but I can totally understand why we would want them to work with nonmembers too.
What really puzzles me are the /cpublic iron door, /cpublic iron trapdoor and cake: it couldn't make any sense to me, so I filed a bug report: http://www.ecocitycraft.com/forum/threads/interact-flag-bug.137519/

Spoiler: UPDATE 2015-10-28

It seems that the strange behaviour of /cpublic iron (trap)doors comes from the interaction between lwc and worldguard plugins:

Therefore it's unlikely this will ever be fixed (plus, JamieSinn thinks it's intended behaviour - I still think it's weird, but I get why it happens).

Did I miss any block/entity/whatever? Do you think I did something wrong? Please leave feedback!

Thank you.

 

This is weird. Have you tried this recently? It might be that meanwhile something changed. I could see with my very eyes the nonmembers using the boat, so it couldn't be a client glitch.

 

Yes. I specifically created a mini pond in one of my plots and placed the boat in there: the user [EDIT] COULD [/EDIT] ride it.
Wait: might it be that you remember trying to place the boat? Because for that you need to be a member of the region period, regardless of the interaction flag.

[EDIT: sorry, I miswrote. I intended to write could, but I wrote couldn't.]

 

I miswrote: I intended to write could (fixed now).

 

Posted update on /cpublic iron (trap)doors (see main post)

 

Maybe I just want members to not be able to break it?

But let's accept your point. Say I want players to be able to open a door: you say I can just /cpublic it. But... this time the door is wooden. And I cannot possibly allow them to.
I'm not saying that /cpublic iron door shouldn't be allowed: I'm just saying that, either both iron and wooden doors are, or none. Otherwise, it's not coherent.
And it's weird that something that in vanilla is more restrictive than wooden (iron works only with redstone), when it comes to lwc and worldguard becomes more permissive.