Just sand those louvers; a tool something like this one
Profile sander will help. It's really not that hard once you get into it. Teak sands different than most woods because of the oil in it, so use 60-80 grit to keep it from clogging sandpaper. I had to remove the teak frame and screen on inside of my door to get to back side of louvers, but it came out great.
This is my door about a year after I did it.
I used Honey Teak on the door, frame, and steering wheel hub, but have lately been using Epifanes Wood Gloss on the rest of my teak, since Power Boat Reports testing showed that it was the most durable varnish. It's also one part instead of 3 parts like Honey Teak, so easier to use. Just follow directions, apply about 6-7 coats, and you'll find that it's much easier to maintain than using oil. I used to use the harsh 2-part teak cleaners and then oil it, but the oil oxidizes and turns dark in just a few months, and the harsh cleaners destroy the soft part of the grain, leaving it very rough. It's been about 4 years since I did mine; door frame will need some touch up soon, but all the Epifanes-coated stuff still looks good.