Short answer. Not on his own. Article 41 says that after the issuance of an employment permit the alien shall not transfer to another job or change his employer without prior approval of the Secretary of Labor. The permit attaches to a particular post and employer, not to the person holding it.
What the law says
After the issuance of an employment permit, the alien shall not transfer to another job or change his employer without prior approval of the Secretary of Labor.
Labor Code, Article 41 — Alien Cannot Change Employer. Read the full provision →
Two separate things are restricted
The prohibition reads: After the issuance of an employment permit, the alien shall not transfer to another job or change his employer without prior approval of the Secretary of Labor. Read the two limbs apart. Changing employer is the obvious one. But transferring to another job is restricted as well, and that catches internal moves — the same company promoting a foreign employee into a different function, or shifting him to a role the permit was never assessed against. Companies that would never dream of skipping a permit for a new hire do this routinely by reorganisation.
"Prior approval" means before, not afterwards
The word doing the work is prior. The article does not describe a notification, and it does not describe something that can be regularised once the move has happened. It contemplates approval preceding the transfer. That is why the sequencing of a resignation matters so much in practice: a foreign employee who resigns first and arranges the paperwork second has, for the intervening period, no permit covering what he is doing. Timing, not intention, is what usually creates the problem in these cases.
What follows a move made without it
The same article provides that a non-resident alien who takes up employment in violation of this Title and its implementing rules is punished under the Labor Code's penal provisions, and then adds a second consequence: In addition, the alien worker shall be subject to deportation after service of his sentence. The two stack rather than substitute. That is the reason a transfer question is worth treating as serious even when everyone involved regards it as a formality — the exposure lands on the worker personally, and it does not end when the employment does.
If a move is being planned
Deal with the approval before the resignation letter, the new contract and the announced start date, because all three are hard to unwind. The receiving company should ask to see the existing permit and what post it names, rather than assume that a foreigner already working in the country is free to be hired. And where the change is internal, look at what the permit actually describes: if the new duties are not the ones it was granted for, this article is engaged even though nobody has changed companies.