I have bounding boxes detected in RGB images, but no information about the camera's intrinsic properties. Using deep learning (preferably) can I find the distance to the object from the camera in real world (meters or cm)? If I find the camera pose (using PoseNet) does that help?
Without going into detail, to answer your question, yes you can. That is, the camera images do contain information about the distance of objects, but only in conjunction with information from the environment which is typically embedded in a model - for example a model about objects, how they appear and their sizes.
Essentially, it would have to work similar to how humans determine the distance to objects that are beyond the range where the separation of our eyes can utilize stereo cues. That is, we estimate how far away a familiar object is using its apparent size. If you see a car with a small apparent size, you assume it is far away because you know the typical size of a car.
This obviously requires an object recognition model that also provides actual size information that can be used in conjunction with your bounding-box apparent size to determine the distance.
Keep in mind that, not only will it only be as good as the object recognition model, but it can be fooled (- is it a real car in the distance or a toy car closer?).