Can he take advantage of a mistake by the phone company to make calls for free?

Praise be to Allah.

What you mention has to do with a technical error made by the company. The basic principle is that it isn’t permissible to take the wealth of a Muslim without his consent. So long as the company has not announced that it would overlook this mistake, it is not permissible to take advantage of it. Rather you should have contacted the company and told them about it. The Prophet (peace and blessings of Allaah be upon him) said: “No one of you believes until he loves for his brother what he loves for himself.” Narrated by al-Bukhaari (13) and Muslim (45). 

And Allaah knows best.