If you have any problems with your knees, ankles, feet, hips, it is better to weigh less when you walk, as each pound adds even more stress to those joints. If you have no aches or pains, I don't recommend against using a weighted vest, but I think there are better (or more useful) ways of achieving the same goals.
Theory #1: Burn More CaloriesThe more you weigh, the more calories you burn per mile. This is true, but the difference is small for any amount of weight you could comfortably add to your body. I'd rather not mess with adding weight to my body, I'd prefer to walk an extra minute to achieve the same effect. I usually wear a hydration backpack, carrying a couple of pounds of water and a couple of pounds of backpack, cell phone, wallet, etc. I don't do this to burn more calories, I carry it to have water and essentials with me on a walk of more than 30 minutes. However, a backpack has disadvantages -- the straps may be uncomfortable, and all of the weight is on your back unless you use the chest strap and belly strap to evenly distribute the weight to your hips rather than your shoulders.
Theory #2: Speed TrainingA weighted vest is used in many sports for speed training. If you train with the vest, then compete without it, you will go faster. If you have already trained fully for speed and distance and achieved the best results you can, then this may be a way to get further increases in speed. But if you still have basic training to do for speed and distance, concentrate on that without adding weight.
How to Walk Faster
Walking Posture and Stride