Putting an ESP8266 to sleep via either of two signals
Figure 1. Schottky diode IV curves. Source: CMLSH05-10DA datasheet.

If you want to use diodes for the OR gate you could consider Schottky diodes which have a lower forward voltage drop. If you were to aim for a 350 mV drop at 25°C you should restrict the current to 0.3 mA. \$ R = \frac {V}{I} = \frac {2.95}{0.3} = 10 \; kΩ \$.



As an aside, does anyone know why the 25°C curve is between 125°C and 40°C?