MAX7221 experiencing 40C temperature rise above ambient
You have set the segment current at the maximum recommended, which results in 40mA/segment. So the chip is drawing as much as 320mA (depending on how many segments are on). The red LEDs will drop about 2.1V at 40mA according to the data sheet. 



So, if all the LEDs were on, the maximum power dissipation in the chip would be (5-2.1)*0.320 = 0.93W. If you are using the SO package, the thermal resistance is 85°C/W so the die will be running at about 105°C with a 25°C ambient (assuming nominal power supply voltage etc.).



As the temperature rises the LED drop will decrease and the supply voltage could be on the high side so this is only a conservative calculation if you assume that all the segments will likely never be on at once. 

The root cause is P= E*I.