Seven Segment problems,voltage and current
Choosing the resistor correctly will stop your LED burning out and, by the looks of the specification this display is very lightweight when it comes to current. It says: -


Absolute maximum current is 20mA
At 20mA the LED segment may develop 1.8 volts across it


If you worked to these limits, and assumed your 5V power supply might be 5.5V, the value of resistor that protects the LED will be: -

\$ \dfrac{5.5V - 1.8V}{0.02A}\$ = 185 ohms.

This means your resistor should not be less than 185 ohms to produce 20mA through the LED with a little headroom on power supply variations and LED forward voltage drop variations BUT this is still only guaranteeing the current is 20mA and THIS value is an absolute maximum rating. Based on this, I'd probably add 50% to the resistor value and pick a 270 ohm resistor.