Will dawn take out the brown grass in the lawn?

Not necessarily. It is first important to determine WHY you have brown grass in the first place. There are HUNDREDS of reasons why grass will go brown. From disease, to drought, to insects.

If the reason is an insect pest such as white grubs (in cool climate) or mole crickets (in warm climate) then detergent will do nothing to help your lawn. (Oddly enough though, Dawn detergent is used to flush out insects in an effort to diagnose a pest problem!)

If you have a disease problem which has killed the crown of the plant, then no amount of water will bring it back to life.

However, if the reason for your brown spot is because your soil has gone hydrophobic (ie. it will not TAKE IN water) and therefore the lawn has dried out, a solution of 1 part dawn to 20 parts water will be enough to lower the surface tension of the water and allow it to penetrate the soil and hopefully rehydrate the roots. If you have Kentucky Bluegrass, or some other "Drough Avoiding" species of grasss, it should come back to life within a few days of being rehydrated.

However, you should know that if the problem has gotten so bad that the grass has died as a result of the drought stress, you are best to re-seed or re-sod the area and start from scratch!