Practically speaking, yes, but not to the degree many people believe and certainly not enough to keep anyone from losing weight. Data reveals that for every decade after age 25, your resting metabolic rate (the amount of calories you burn while sitting quietly--RMR) slows down approximately 2%. While it is true that RMR does decline, this would only be 30 calories per day for someone whose RMR is 1500 calories per day.
Why does RMR decline? Loss of muscle and hormonal and neurological factors all probably play a role. One must also keep in mind that the slight decrease is usually offset by the increase in weight that happens to most adults as they age (the heavier you are the more calories you burn).
A sedentary lifestyle is usually associated with a gradual loss of lean body mass (LBM) and an increase in fat mass. This is why, though they may not gain much weight over time, some people look and feel considerably flabbier. They have negatively impacted their body composition (muscle-to-fat ratio). Staying active and making consistent, healthy food choices can keep this metabolic slow down at bay.
And most importantly, simply adding a little movement throughout the day will more than offset any age-related slow down in metabolism.