In trying to understand 3D printers, I have watched some YouTube videos where the crafters make items with hinges. That in itself blows my mind. It is hard to grasp how something with moving parts can be printed. But specifically I am wondering if the concept can be extended to food printers to make, for example, a sugar or chocolate telescoping lollipop (sucker)?
I don't know much about food printers. But at the very least, you should be able to print similar moving parts like hinges as you would with a standard FDM printer.
When printing things like hinges with a single extruder printer, the machine will typically be configured to include support structures. This is basically very thin scaffolding that higher layers of the print can sit on.
As Tom van der Zanden pointed out, though, this highly depends on the food medium you wish to use. With plastic, the material is heated to a less than liquid point, usually allowing the material to "bridge" across gaps. If your food medium is too thin (or close to liquid) you will not be able to effectively print supports as the medium will fall between the gaps.
If you are able to effectively print the supports for overhanging features, you should be able to knock off the supports with a small knife or toothpick. If it's a moving part, sometimes "shimmying" the part will knock the supports off.