Input current specifications of logic inputs have nothing to do with "current rating". They are an information about expectable current of this high impedance inputs. There's no risk of overloading the inputs as long the input voltage is in the permitted range, in this case if TLP350 and IR2110 share the same power supply.
TLP350 can overheat if the LED input current isn't properly limited or if the switching frequency is too high. And of course if the supply voltage limits are exceeded.
I also don't understand why you are using TLP350 with IR2110. TLP350 has sufficient gate drive capability, it won't be increased by adding an IR2110. You would rather use a fast logic level optocoupler in front of IR2110 or TLP350 stand alone.