What is the diode for in this 555 circuit?
The diode improves the stability of the current source with temperature changes.

Imagine the Zener voltage is stable with temperature, then without the diode, the current will increase by about 2mV/Re = 4uA for every degree C. The diode forward voltage drops by about 2mV every degree C, so it mostly compensates for the transistor Vbe drop.

They are operated at different currents, so the compensation is not perfect, and the Zener will have some tempco (not much because it will be close to the optimum voltage for Zener tempco) but this will improve overall stability by perhaps 5:1 or 10:1. Over a range of, say, -20'C to +80'C it might improve the stability from 5% to 1%, where the capacitor tempco will dominate. It's a reasonable thing to do if you care about the frequency. I once used a similar circuit without the compensation in a dual role to generate an ADC ramp (homemade 15 bit ADC in this product) and to measure board temperature.