Water the area immediately to dilute the dog urine before it has time to damage the grass. If neighborhood dogs are the issue, water daily to help keep any urine in the lawn diluted so it doesn't cause damage.
Avoid adding nitrogen-rich or urea-based fertilizers to the affected area during your normal fertilizer routine. These fertilizers will just add to the problem and cause further burning or grass death.
Rake the damaged area to remove the dead grass and thatch, revealing some of the soil beneath. Spread fresh grass seed over the location, using the same type of seed as the surrounding lawn grass. Keep the seed bed moist until the seeds germinate and the grass establishes.
Remove the entire patch if the grass is completely dead. Dig up the grass and up to 1 inch of the topsoil. Cut a sod piece to fit the bare spot and set it in place. Keep the sod evenly moist until the roots establish in the soil beneath.