In a particular DS9 episode, the Female Changeling tells Odo something along the lines of, "When you're a changeling, you're timeless."
Does this statement mean that changelings don't age, and cannot die of natural causes?
"In a sense", to quote an expert. It depends how you look at it.
In DS9 "Behind the Lines" Odo gets some insights on the nature of the Changelinghood:
So, in a sense, there is only one Founder: The great link. As long as it exists, all "drops" of individual Changelings, that constitutes this Uber-Founder, exist. So at least in that respect, Changelings are immortal.
In DS9 "Children of time", we learn that individual Changelings have a very great life-expectancy. In a colony founded by the crew of the crashed Defiant two hundred years ago (there is time-travel involved!), the descendants of the crew live alongside Odo, who is over 200 years old. He is not only still alive but seems to be in excellent shape.