First of all, the fact that a particular image is claimed to be "in the public domain" does not make that true. Very few images first published after 1964 in the US are in the public domain in the US, unless they are so simple that they were not copyrightable in the first place (not original enough). (Images older than that may have lost protection by failure to renew copyright, although that takes some work to prove in a particular case. Images published before 1923 will normally be in the public domain under US law.)
The "I (heart) NY" logo is primarily protected by trademark, not copyright, and the rules there are different. You cannot use that logo, or a similar logo, to sell or advertise a product or service, or in general in the course of trade, without permission from the trademark owner. Use not in trade is not governed by trademark protections. For example a book about trends in advertising, or one about landmarks and tourist destinations, might reproduce images of the logo. The fact that such a book is itself sold commercially does not mean that every image within it is "used in trade".
However, if you want to reproduce an existing image, such as the postcard, you must be concerned with the copyright on that image. You would need to determine who owned the copyright, and secure permission, or determine when and how it was placed in the public domain, or under a license that you could use. You may not safely assume that any image is in the public domain without checking. That an image was posted on the internet does NOT mean that it is in the public domain, or is legal to reproduce.
If you yourself take a photo of a publicly posted sign that includes the logo, there is probably no copyright issue. However, if you are going to publish a book, you would do well to consult a lawyer with your very specific details.