So Chlorine Trifluoride is a very dangerous chemical known to burn through things like concrete and asbestos. It's so bad that the Nazis, THE NAZIS, decided it was too dangerous to use. So I need an organic substance that takes forever for this compound to burn through (3 hours to a day) that is about as thick as your skin. I am willing to take softer science answers (like silicon or germanium bonded to carbon chemistry like some microorganisms have done) but I would prefer to stay as close to mammalian biochemistry as possible.
Almost exactly the same question was asked here. Linked is my answer.
The answer: make the gear out of calcium fluoride, or fluorospar. Chlorine trifluoride is caustic because it fluorinates anything that can be fluorinated which is just about anything. But calcium fluoride already has all the fluoride it can have. It is full of fluoride and no more can be added. Also, it is a stable crystalline substance.
Make your vessels out of fluorspar. Bonus: it is pretty.
Unfortunately you ask the impossible. Chlorine trifluoride will eat through any organic material, fluorinating the carbons. Even teflon is not completely safe from it, because the reagent will attack the carbon-carbon bonds.
You need a substance that is both completely fluorinated and mechanically stable, which pretty much restricts you to solid metal fluorides, calcium fluoride, etc..
You might have your organism secrete calcium fluoride or copper fluoride, but then you wouldn't have a mechanism to produce those organically. Remember that the final product must be completely anhydrous, or chlorine trifluoride will burn through.
If you read this: https://www.cdc.gov/niosh/ipcsneng/neng0656.html Practically nothing. Any biological substance with water in would make for a horrifying show. Maaaybe if you can biologically produce large amounts of Carbon like Graphene, which contains no water, you might be slightly safe from the burning (but not of the vapour as it is now boiling and getting everywhere). But after reading about this substance I'm willing to bet that pure Carbon is somehow also going to do something horrible when in contact with chlorine trifluoride. In fact, most of the time I see "fluoride" somewhere in a molecule name it's usually something naaaasty.