Patch for a rip in the lining of a down jacket
When i worked at a tailor shop, we had single side fusible backing to repair the liners of the tuxedos. It came in black or white, was placed on the inside of the tear, ironed into place and then, if it was a large tear, hand stitched.
    The easiest thing to do in this situation if you don't need to stock up on this type of product, is to take the jacket to a dry cleaner or tailor shop and get a quote on the repair.


If you want to buy the interfacing, look into Pellon Shape-Flex Interfacing. Instructions are below the image.


Quickly looking online, the example above is the closest thing I can find to what I remember using without actually touching it. It was malleable and had a knit texture to it, this example says woven, so the texture may be different. You'll want to find something that matches your repair fabric. The fabric shop may sell facing by the foot or yard. (editorial - You really do not need an entire bolt unless you're going into the repair business. A little goes a very long way.) 

Instructions: You only need to cut as much as you need to hide the tear. The facing goes inside the tear, adhesive side touching the lining. Then iron with a press cloth to protect the polyester. As stated above, if it is a large area, it's good to hand stitch the tear for extra security. 

For a hole, place the facing inside the hole, adhesive side against the lining, then place a small piece of matching fabric over the facing, use a press cloth when ironing. Then hand stitch if necessary.