Not sure what to do with the brown, burnt layer of my fence trees: what will happen if I do nothing? Should I trim all the brown parts?
Makes no difference whether you trim it off or not in respect to the tree dying or not dying - what's dead will remain dead, and eventually start to fall from the tree. The only advantage to cutting it back is its good to cut back to live wood, and it looks a lot better without all the dead brown stuff. Not sure what the conifer plant is that comprises the hedge, but it looks like something you trim regularly, in which case, it's probably worth trimming back the dead parts. If its due for clipping any time in the next couple of months, you could just do it then. There seems to be some more extensive charring in top right of the picture though, that's not just superficial.
This is chemical damage. What have you sprayed recently anywhere on your property? To have such a marked difference between DEAD and HEALTHY is indicative of chemical/man made damage. Yes, prune it but we need to determine what caused this! Looks like someone sprayed your trees with an acid or paraquat. Luckily this isn't systemic. This is some weird overspray that literally has burned the foliage on these oh so healthy arborvitae! What has happened withing the last week maybe two?
When you prune, clean your hedgers with alcohol. This is not some disease but transferring whatever chemical has burned this foliage is not a good idea. Also prune so that the width of the bottom is the widest point. The top should be narrower! Still, your arborvitaes were so dang healthy! You must surely be able to remember something being done just recently.
Neighbors? City? What a huge bummer! Top these trees (arborvitae or western cedar variety...'virescens'?) and angle the sides like a pyramid? only not that steep to the tops. Gotta find out what the heck happened here! This is a chemical spray not something that the roots took up!