Yes, remembering dreams can make you "live longer" so to speak.
This applies a couple different abstract concepts, so bear with me.
First, the idea that all a person knows and experiences is dependent on perception and memory. The world we live in and our very selves are all known to us solely based on memory and incoming data. Without memory, a person would not be who they were. In essence, memory is life. The more experiences a person has while living, essentially the more they've taken out of life. Which is what living longer is all about, experiencing more while living and thus making more memories.
Now to the dreaming part. It is a fact that people dream every time they sleep, whether they remember it or not. Dreams, although sometimes distorted or meaningless, are real experiences. Considering the nature of reality, a dream is technically just as real as daily life. It can be meaningful, crazy, fun, sad, anything. Therefore when a dream is remembered, it is like any other experience, it is an additional memory, additional knowledge and experience gained while aware in this reality we live in.
So think about it, when a person is not remembering their dreams, they are missing out on thousands of hours of experience and memories throughout their life. Remembering ones dreams has a similar effect to living longer!
And to finalize the importance of this, it is not difficult at all to start remembering one's dreams! A simple dream journal and the desire to remember them is all it takes. If you think you don't have dreams, just tell yourself before falling asleep "I will awake and remember my dreams" and then be ready to write them down when you do. You will remember something!
So what are everyone's thoughts on this?
And to drag this on even longer (sorry), I feel it's important to note lucid dreaming too. It is possible for anyone to learn how to have somewhat frequent lucid dreams, dreams in which a person is aware they are dreaming and can then operate in the dreamworld in a more realistic manner. What a concept! Sleeping while still living.
The wikipedia page describes it well