OLEDs tend to be fairly low power since they don't have a backlight, but they do require high voltage to drive the OLED segments. For this reason there is a boost converter on the back of the OLED. There's also a 3.3V regulator. However, we've found that it can be very sensitive to noisy 5V supplies such as that on an Arduino so try to run it off the 3.3V line which is filtered and cleaner. The power usage will vary with how many pixels are lit, the maximum is around 25mA.
Of course, the microSD card is another matter - and may draw up to 150mA or more during writes (check the OEM document for the card to understand current usage!)