Is it safe to directly connect an infrared led to an Arduino?
According to your link "The KY-005 Infrared Transmitter Module consists of just a 5mm IR LED.". This is confirmed in this YouTube video (image below is from the video).



Infrared is not visible to the human eye, but you can check to see if it is emitting
light using the camera in a cell phone or laptop computer.

Connecting it directly to the Arduino I/O pin produces the most light, but relies on the internal resistance of the port to limit current. This is safe provided no other pins are driving heavy loads (ATmega328 absolute maximum current for all MCU pins combined is 200mA). It should still work with a resistor, but with shorter range. With 220&ohm; it should draw about 15mA, which is around half the current it draws with a direct connection. 

Note: do not connect the 'signal' input directly to +5V, as it will burn out the LED!