In Ratatouille, when Linguini took over Gusteau's by proving he inherited the restaurant, Skinner seemingly lost his job. Was there legal precedent for Skinner to lose his job; would Skinner hiding the fact that Linguini was Gusteau's son be legitimate grounds for dismissal, or would the fact that there was no station left for Skinner (as his former station as sous chef was replaced) be enough (redundancy)?
It's been a while since I've seen the film (and what I remember of it is the ending), but if I recall, Skinner was conspiring to conceal evidence that the restaurant was legally Linguini's so that Skinner could become the legal owner under Gusteau's will. As such, under French Law, an employer may fire an employee for disciplinary reasons that fall into three categories: "Faute Simple", "Faute Grave", "Faute Lourde". In likelihood, since legally Skinner was attempting to steal from the owner of the restaurant, this would likely fall under "Faute Lourde." This is defined as when "when an employee intentionally and willfully attempts to harm the employer or other employees. In this case, the employer must demonstrate intentionality." If not, it would certainly be Faute Grave, which is the same as "Faute Lourde" although the reason does not have to show the intent was malicious, but that it simply harmed either the employer or an employee. At best, Linguini was within his right to fire Skinner without a notice period or pay for the unintentional concealment of his ownership of Gusteau's, though if I recall, Skinner's malicious intention was clear to Linguini by the time he was declared the legal owner. (source)
Ratatouille is set in France, so I'll defer to others on application of French law.
But in the united-states, the general rule is that an employer may terminate any employee, for any reason or for no reason at all.
There are of course exceptions. You can't fire employees because of their race, religion, sex, and so forth. You can't just fire employees who have a contract for a set period of time or that provides for progressive discipline. And an employee's right to continued employment in a government job may be protected by the Due Process Clause.
But beyond those rules, an employer can generally be entirely arbitrary in who they hire and fire. An employer can promote you over someone with far more experience simply because you bring doughnuts to the office on Mondays. And the employer can fire you the very next day simply because he thought your shoes were ugly.
My understanding, though, is that French employment law is quite a bit more protective of workers.
If we accept the terms of the inheritance as legal, and if we accept that Skinner and Linguini are both designated as heir in the inheritance, Skinner knowing that Linguini is entitled to the restaurant but withholding that information would likely be recel successoral (stealing inheritance basically, Code Civil Art 778).
There's precedent for infractions committed outside of work to be cause for termination. The reason can be that it happened on the workplace, even outside of work hours, or that it is linked in some way to work. So Skinner's tort could become ground for a faute as discussed in hszmv's answer.
Licenciement pour motif économique (Code du travail Art L1233-3) is a possibility but it still requires a cause, the "motif économique". His position would have to be deleted and his functions distributed to various remaining employees, or his position would have to be transformed and it would have to be impossible to train him for it. Simply giving his job to someone else isn't reason enough (as explained here).
All the above presumes Skinner is employed in the first place and that his employer is Linguini, which is dubious. Linguini can't become the owner of the restaurant or Skinner's employer through sheer wishful thinking. It's been a while, so I can't recall the exact circumstances of Skinner's dismissal. Wikipedia says "Linguini [...] forces Skinner out", so this reinforces the idea Skinner wasn't actually fired.
"Forcing" someone out in these circumstances might be a form of chantage (blackmail; Code Pénal Art 312-10), forcing Skinner to give up the business or else he'll reveal his malfeasance. It would be unlikely to be harcèlement moral (bullying; Code Pénal Art 222-33-2) as it's a single occurence and not repeated.
There are legal processes to contest the inheritance and the ownership of the restaurant. Threats isn't one of them. Both Skinner's and Linguini's action would be cause for action.