Is it bigger than a bread box? In any case, we'll start by assuming that you're talking about something made by humans, not a tree, or a waterfall, and that it is publicly viewable. Assuming that's the case, here are the rules.
Copyright. If the monument is a creative work like a statue and it's not that old -- pictures of it weren't published before 1923 -- then the person who created it (or the person who hired the creator) has copyright and can control reproductions of the work. So, if the non profit is the copyright owner, you may be in luck. Note, fair use is a possible defense when works are publicly viewable -- it all depends on context (is the work the heart of the picture) and use (is it transformative)?
Trademark. The monument can serve as your non-profit's trademark but that doesn't mean you can stop others from selling pictures of it. Your rights to the mark are limited to the classes of goods or services offered by your non-profit and most likely you would probably need to demonstrate that consumers associate the appearance of the monument with your organization. So, for example, Liberty Mutual was once able to stop others from using the Statue of Liberty to sell insurance (the mark has expired), and Paris Accessories can claim exclusive rights to use the Eiffel Tower to sell handkerchiefs. But courts have been reluctant to claim that the owner of a trademark of a distinctive building for example, can stop photographers. As one court noted, such photos don't confuse consumers as to the source of goods, they're just perceived as a photo of a well-known and accessible public landmark. So, we think your course of action is an uphill battle that will require consultation with a trademark attorney.