You need to assess whether they can handle the rigors of the show, as well as whether or not you can break someon else in at short notice. Your first obligation is of course to the show so if they truly can't handle it, you may need to look at it honestly and replace them. However, you don't want to alienate people with whom you may want to or have to work with again in the future. Tact is critical. Remember that if a person has been given a job to do for which they are not sufficiently skilled or trained, that is a managerial oversight. Whether you keep them or not, though you need to make sure that they feel supported and not condemned.
If you plan to keep them on the job, you need to make sure that they have every technical support (charts, prep sheets for cues--whatever) that you can afford them. Are there ways to streamline what they have to do? Are the expectations placed on them reasonable given the factors of time, equipment, etc? You need to ask first whether you have done all that you should to facilitate them.
The tantrums are self-indulgent, to be sure, but are probably the only way they know to handle the embarrassment of having screwed up. If they are teachable, then they will warm to your leadership if they feel that you are behind them and trying to help them improve. If they can't find the humility to improve, then you have a hard decision to make, I'm afraid.