If I'm on a calorie deficit and trying to burn fat, at what point do I start burning muscle instead of fat? How can I be sure that I'm eating enough to minimize muscle loss, but remain on a calorie deficit so that I can burn fat?
If you are "actually" overweight, meaning that your body considers itself to have excess fat, then you can restrict calories significantly as long as you maintain a healthy lifestyle of exercise and can do so without becoming exhausted or excessively hungry. Some people can reduce intake into the low hundreds (300-500) or even stop eating completely for a period of time and stay active and are healthy, while others have to stay above 1000-1200 to maintain an active lifestyle.
Personally, when I was on a weight loss plan, I was able to go for a period of time eating 3-4 small meals (~500 calories per meal) each week. This only lasted for a few weeks, and during that time I lost an average of a pound each day. During that time, I walked 4-5 miles every day and remained almost as active as normal, with only very slight changes.
Your body is intelligent enough to start burning the stuff it needs least first, and will not start to burn needed/used muscle until you are down to your lowest healthy body fat percentage. If you sit on the couch and don't eat, your body will consider muscle fair game since it isn't being used.
Calorie reduction will also cause dehydration, and the further you reduce, the more severe the dehydration becomes. Be certain to drink all the pure, plain, unaltered water (I stress pure water) that you need. Drink water any time you feel thirsty.
Talk to your doctor before you decide to do any diet, especially any form of extreme calorie reduction. Some doctors refuse to recommend extreme calorie reduction for patients, while others are fine with it, but at least ask about the risk specifically presented to your health, as your doctor will know your health history best.
Be absolutely certain that you do not continue a calorie restrictive diet past the point of reaching your minimum healthy weight, or you will begin burning muscle. The muscle you will burn includes your internal organs such as your heart. Taking muscle from your organs is extremely hazardous to your health and can lead to long term or permanent damage.