Will I burn existing body fat faster by trying to eat close to no fat in my diet?

no, it doesn't work that way.

Thanks, I didn't think it did   but instead of me never asking I thought I would ask.

Not that I recommend it and everyone is going to go crazy about me saying this... but if you cut out carbs you will put your body in ketosis and it will burn the fat since it has no carbs to burn and/or storm.  Ketosis happens when you consume less than 70g of carbs a day, however, most people jump start it by doing less than 20g of carbs a day... i.e. the atkins diet...

I assume Ketosis is the name of your local hospital?

You need some fat in your diet.  Some fats are better than others though, so if you want to make a change, try swapping out some of the not-so-good fats for better ones.

Mostly, just keep training and living a healthy lifestyle, your body comp will change over time.

Not only doesn't it work that way, but your body really needs fat. In fact, the reason HDL is considered "healthy" cholesterol is that it's essential to remove LDL and VLDL (bad cholesterol). Fat is required for cardiac and brain function, hormonal control, assimilation of many vitamins and minerals, etc...get the picture? NEVER cut your fat below the recommended intake--just stick to healthy fats and cut your intake of anything that's solid at room temperature, as a rule of thumb.

As for burning fat, triathlon training is a great place to start :-) From a biochemical/physiological standpoint, there is some truth to the fact that you'll burn more fat by doing cardio early in the morning and without eating first. Unless you're the type who normally does this, I wouldn't. I usually have a banana or something small to fuel a tough morning workout, then eat a good breakfast after that.

The best advice is pretty standard stuff--ramp up your cardio and watch your calories. Adding intervals to any of your workouts will up your calorie burn, improve your performance, and (evidence suggests) burn more fat. Don't cut calories drastically. Just use an online calorie calculator to determine exactly how many calories you need to maintain your current weight, factoring in exercise, then subtract 500/day to lose one pound per week. Try to "eat clean," making most of your meals lean proteins, complex carbs, fruits & veggies. Splurge on pizza or burgers or whatever you like once or twice a week, but keep it reasonable--two slices, not half a meat lovers pie!

Besides cardio and diet, don't forget weight training. Endurance training and calorie counting alone will help you lose weight, but that includes muscle mass. Many thin people (including endurance athletes who ignore this) actually have a high body fat percentage, even if they're not fat. Again, interval training and weight training will help you build muscle and lose fat overall.

I've been able to lose about 19 pounds (mostly fat I'm sure) in the last several months by changing my diet based on the Paleo Diet for Athletes.  Basically, it's eating lots of fruits, vegetables, nuts, and lean meats.  I cut out grains, dairy, bad fats, and high sugary foods, basically anything that's processed.  I'm down to the weight I was in college.  I went from 177 to 158 today.  So yes, I did cut out almost all bad fats and replaced them with good fats, but I also cut calories in that I cut out lots of foods from my grocery list.

My aunt was told to go on a low fat diet. She thought that if low fat is good, no fat is better. It killed her. Her body started burning muscle tissue and it ate her heart up. She wasn't training as much as you (treadmill about 30 minutes a day) and had a lot more body fat than you to burn off. I wouldn't recommend a no fat diet.

First rule of thermodynamics - burn more calories than you eat.  I know it sounds simple and it is.  Not easy, but simple.
Remove anything that is not natural - the further from 'straight from the ground' generally the worse it is for you.  Eat fresh food, don't eat out, drink lots of water, avoid alcohol and soda.

What?  Actually keto based diets are great for those who are not endurance athletes.  I actually think that weights plus a keto based diet w/ low intensity cardio for short periods of time during the off season for triathletes would work fine.  Check out books by people like Shelby Starnes if you go this route.  His Guide to Very Low Carbohydrate Dieting is very good.