I'll probably be more upset the day The NY Times stops publishing the print edition.
DMOZ is without a doubt an icon that has jump-started web classification with hundreds, if not thousands of entrepreneurs rushing to run their own Yahoo!, whether based on DMOZ api or their own. It has served the web well. In terms of SEO impact, I don't think there will be a meaningful effect, since the game has changed so much that a tweet is probably of more value to your business than a listing at this classic catalog.
In a metaphorical way, DMOZ is the newspaper of today's news media.
What is your take on DMOZ shutting down? I applied to be an editor there years ago, but was rejected with no explanation. I did manage to get a few companies listed there, but it wasn't easy.
I swear they were around longer than since '98. I remember submitting websites back in the day and while there was rarely ever traffic from it, it was a great source of SEO power.
As we moved into the 2010+, their SEO influence was less and less and really for the past couple of years, they've been a non-factor.
It's been an interesting journey for them, and I'm surprised someone like Verizon didn't pick it up (in addition to their Yahoo acquisition).
All good things must come to an end.