The vinegar will persist in those spots as a lower pH soil. Vinegar is used as an herbicide (a much higher percentage than what we use in salads) and prevents anything from being able to grow...great for driveways for instance. All one has to do is add lime to get it back into the pH range to grow plants again.
I would dig out the dead grass and soil, replace with fresh soil, test your lawn for pH. If it is below 6.5, use lime to raise the entire lawn's pH to between 6.5 and 7.0. If you haven't aerated this year, now would be a good time. Leave the plugs on your lawn. Use a hand spreader and reseed with a grass mix that says 'zero weed seed' on the grass seed list.
Fertilize (if you haven't fertilized for 2-3 months) with Dr. Earth's Lawn Fertilizer. Follow the directions. Use your hand spreader, not your hand to apply fertilizer. This stuff is worth the money, takes longer to show and lasts longer than the regular fast response inorganic fertilizers.
Mow no lower than 3"! This will prevent weeds from being able to germinate as the sunlight can't reach the soil. Also if your seed mix is like the Pacific Northwest's the grass plants have large root systems, genetically. Without enough top growth (3" minimum), the plant is not able to make enough food to support that root system and will be stressed. Stressed grass can not compete with weeds successfully.
Salt will be washed away. Don't worry about this. While the seeds are germinating (about 2 weeks after you over-seed) DO keep the surface of the soil moist, all day, every day. After your lawn has filled back in, water deeply and do not water again until you can step on the lawn and the grass doesn't spring back. This is training your grass to get deep roots and become drought tolerant.
UPDATE: Just thought I'd mention treating just the spots will cause the 'spots' to persist longer. That is why my opinion includes treating the entire lawn to prevent the original problem.
Boiling water or boiled liquid will kill the top part of the plant. The vinegar changes the pH drastically and the salt I have no idea why they put that in your recipe. Where did you get that recipe? Did they mean for this to kill dandelions in a lawn? I'm interested to see their thought process, truly.