If you make one of the circuits described on this site and it lies on the table with none of its wires touching, you probably won't need any insulation.
If you are going to use your LEDs in a practical manner, they will probably take a bit of a bashing and inevitably bare wires will touch. If this short circuit means the current can bypass the resistor, you will ruin all your LEDs.
If the short-circuit bypasses an LED, the other LEDs will receive too much power and burn out fairly quickly.
Electrical insulation tape is easy to find and pretty cheap. Don't use more than is needed to cover the bare metal, but make sure all parts are covered. More tape can be used to join already-wrapped pieces together.
Other tapes may work just as well ('gaffer' or 'duct' tape for example) but are often painfully difficult to strip off if you make a mistake.
Make sure your circuit is working properly before you add the insulation.