EDIT 4/15/19 -- Understand that it is natural to feel sad and upset over losing something big, even if it is not something you own or have visited personally. The cultural relevance creates a connection that most people feel, whether or not they can articulate it as such. Some resources that may prove helpful in coping with the aftermath:
Coping with Stressful Situations
Emotional First Aid
Helping an Online Friend in Difficult Times
I have not yet seen fundraising for any of the obvious things that are going to need money -- either reconstruction or site clearance, and support for people exposed to toxins such as lead released by the fire -- or other methods of direct support relating to this tragedy, so if anyone else spots those, please include a link.
Check for historic buildings near you and inquire about their fire safety status. Many operate on a shoestring budget and may not have enough funds for things like smoke alarms, annual examination of fire extinguishers, or security to watch for problems. Most have a donation page where you can contribute to the upkeep. Drop them a few bucks and mark it "in memory of the Notre Dame Cathedral" or the like.
It's also a good time to check your fire alarm batteries, fire safety plan at home or at work, etc.
Does your local fire department offer fire safety classes? Do they have the funds to do that for free, or some kind of subsidy for people who need it but can't afford it? Do you have skills to offer fire safety presentations yourself, even if it's just for your own area? Those may be useful responses too.
Enjoy some pictures of the Notre Dame Cathedral. If you have visited there, share pictures or descriptions.
Watch The Hunchback of Notre Dame. It's a breathtaking movie in which the setting really acts like a character.
Write, paint, or otherwise depict Notre Dame in creative works. What is remembered, lives.
EDIT 4/16/19 -- The fire is mostly out now. Despite massive damage to the cathedral and the loss of its spire, the towers remain standing and firefighters have declared it "saved." Reconstruction should be possible.
EDIT 4/16/19 -- Many artists have drawn tributes to the fire of Notre Dame. Check out the repeating motifs! Quasimodo, tears, hearts, the red-white-blue of the French flag, and horizontal striped tops all appear in multiple images. While disasters like this are awful, they do create a great opportunity to see cultural works about the same thing at the same time. I hope that when the cathedral is rebuilt, it will include -- or have added nearby -- an exhibit about the fire featuring some of these passionate pictures. For students of art, it's a valuable lesson to see how artists respond to a major cultural upheaval, and how their expression homes in on a small cluster of closely related symbols. That offers inspiration for others to do likewise, and making art is a very healthy way of dealing with tragedy.