Technically, a piezo element doesn't require a crossover to function properly, as it's already a capacitive load. However, if you want it to SOUND GOOD, then yes, you NEED a crossover.

How can I make my piezo tweeter sound better?

Adding a series resistor greatly improves most uses of piezoelectric tweeters. Adding about a 47 to 50 ohm or so resistor reduces the extreme treble response by about 3 dB at 20 KHz, and by about 2 dB at 16 KHz.

How does a piezo tweeter work?

A diaphragm made of paper or another flexible material is attached to the crystal of a piezoelectric tweeter. This structure amplifies the crystal vibrations and creates vibrations in the surrounding air. These resulting sound waves are interpreted by the ear as high frequency noises.