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?
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"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:
ODO: You haven't told me your name.
FOUNDER: What use would I have for a name?
ODO: To differentiate yourself from the others.
FOUNDER: I don't.
ODO: But you are a separate being, aren't you?
FOUNDER: In a sense.
ODO: When you return to the Link, what will happen to the entity I'm talking to right now?
FOUNDER: The drop becomes the ocean.
ODO: And if you choose to take solid form again?
FOUNDER: The ocean becomes a drop.
ODO: Ah, yes. I think I'm beginning to understand.
FOUNDER: Then you can answer your own question. How many of us are there?
ODO: One. And many. It depends on how you look at it.
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.