There is a well-known axiom that “if it isn’t supposed to move and it does, use duct tape.” While effective, this is generally a bad idea. I do not recommend you have duck tape in your tool box as it’s too easy to fall back on the above.
Gaffer’s tape (as it is also known) is great stuff. It sticks to just about anything, is very strong and waterproof. Unfortunately, because it has such strong adhesive (and construction) it is extremely difficult to remove from anything. As any repair you’ll use it for will be temporary, you’re better off not using duct tape as removing it for a “proper” repair will be difficult and it may damage the equipment you’re trying to repair. It’s great at removing paint (for example) which is not usually desired if used with dental equipment. The adhesive leaves residue on most things and it can be difficult to work with.
If you want to hold something together with gaffer’s tape, I recommend wrapping the item(s) in question first with a layer or two of plastic food wrap. The plastic wrap will provide a barrier making it easy to remove the tape without leaving any residue. You’ll still benefit from the strength of the tape without needing to worry about messy (and difficult) clean up.
Incidentally, I prefer the terms gaffer’s tape or duck tape to duct tape as it’s a common misconception that this tape can be used on duct work. However, it is contrary to building codes to use it on HVAC duct work. Gaffer’s tape was designed to be strong and water proof, but it is not resistant to temperature changes, which tape for duct works needs to be.
Gaffer’s tape can be a nice addition to your tool box, but I find you’re often better off leaving it out to avoid falling back on a temporary “repair” that will be more difficult to undo for a permanent fix.