Can I use a curling iron with this 50 Watt converter?

It's hard to say for sure without seeing a schematic of the device but I assume it isn't designed for a purely resistive load. There are, generally speaking, two types of loads run from AC electricity - resistive and inductive. They have different designs or a device can be designed to run both.

A heating element, whether a curling iron or an electric wall heater, is almost all resistive. The resistance is what causes the heat.

An AC electric motor is an example of an inductive load.

So a hair dryer would have both. A resistive part for the heating element and an inductive part for the blower motor.

If it says not to use it with a curling iron then don't.
 
It specifically says not to use it with curling irons. I don't know why that is but I'd probably heed that warning. Electrocution and severe burns are both things I try to avoid.