Switching 3.5W heater with an ESP8266
For switching "heavier" loads you definitely need a MOSFET, for example IRLZ44N. 3,3V from GPIO is more than enough to turn it fully on.

When it comes to power supply: your resistor will consume 0,625A (I=U/R = 5 V / 8 Ohm), ESP will consume next to nothing, so a 1,5A power supply is enough.