Does cutting off a pineapple's crown shorten its longevity?
It depends on where you cut it - if you are taking off the majority of leaves, but leaving the "fruit" intact, then it should be fine. If you are actually cutting into the fruit, then it will affect how well/long it keeps and how it matures.

Incidentally - it is much better for flavour development to keep fruit like this outside the refrigerator. Also, if you are in a warm country, you can actually cut the crown off a mature pineapple and place it in a pot and let it grow. Over time it will produce a pineapple plant and, provided you look after it enough, even produce a new pineapple fruit.