It seems entities cannot activate tripwires, pressure plates, or buttons, and villagers cannot open doors.  This includes fishing bobbers and items, as well as arrows fired at buttons, but excludes players.

If a player is standing in a pressure plate or tripwire, then summons a mob while the plate or tripwire is activated, it will remain activated as normal until the mob walks away, however it will not re-trigger if said mob walks on to it.  The same is true of buttons pressed while an arrow is embedded in them.  While this is true of all wooden pressure plates and tripwires, this is not true for metal pressure plates, and they will deactivate the moment a player steps off them regardless of if another entity is standing on them, and regardless of if the plate is iron or gold.

Farmers still plant and harvest crops, and zombies can still break doors.  Villagers still move to attempt to open doors, but are unsuccessful.

I have verified, to the best of my ability, that this is a Spigot issue by running the server with Spigot with no plugins and by running Mojang's vanilla .jar.  Everything operated correctly on pure vanilla, however not on plugin-free Spigot.

Steps to reproduce: Make a box around any wooden pressure plate or a tripwire, attach any redstone object to the plate, summon a mob, throw an item, or cast a fishing line in the box.  The plate will not click and the redstone object will not be activated.  If a villager is boxed in with a door, they will be unable to activate the door.