I sat with the exact same question this weekend searching for answers like a mad man and stumbled upon this post. After finally figured it out myself I'm now going back here, creating an account just to answer this if someone else might have the same question.
As many things it's actually pretty simple. You need to install the program exifprobe from which exifgrep is part of.
sudo apt-get install exifprobe
The --info command executes the command in shell so you need exifprobe installed for it to ever work. The build in exifprobe/exifgrep is not working in the shell. With --info you can basically get it to print any command output for example feh --info 'echo "this will be printet on the image"' ./
When exifprobe is installed you can run the following feh command and get the date information you want. The cut command from the example is just to remove some of the text displayed next to the date so you can add that yourself if you want.
feh --info "exifgrep 'DateTimeOriginal' %F"
HOWEVER
I actually ended up with a different solution for myself. You see, exifgrep cannot handle spaces in the path name so if you have a path called "~/Photos/My Favorit Cats/cat.jpg" it won't work, and no matter what i tried to escape the path name it just broke.
Instead i finally found exiftool that you can install by writing
sudo apt-get install exiftool
exiftool handles spaces correctly and can do a lot of cool stuff, take a look at the manual. Most important for you is probably to list all the tagnames of a photo, if you want more that just the date, by writing the following.
exiftool -s "~/Photos/My Favorit Cats/cat.jpg"
The -s is to get the actual tag names, without it you get the human readable names.
Then with this tool i ended up with feh command bellow. Note the dot slash in the end, that just means show all photos in this folder. The cut command is again just to remove the tag name that's in front of the actual date in the string output.
feh --info 'exiftool -DateTimeOriginal %F | cut -d : -f 2-' ./
I hope this is helpful to others searching for the same problem.