Is it safe to use 1/8 watt resistors in this circuit?
P=0.050A x 5V == 250mA


No, the power only takes the voltage dropped into consideration. You need to reduce the supply voltage by the forward voltage of the LED to determine the voltage dropped across the resistor.

But if we assume a forward voltage of 1.2V (typical of an IR LED, one of the few that would use 50mA) then we get a power dissipation of 190mW. So either move up to a 1/4W resistor or use two 1/8W resistors in parallel.