Applying for a license won't have any effect at all on his immigration application.
But the question will be whether he'll be able to apply for a license, given that he's a non-resident. He may be able to based on his driving experience, but that's separate from his immigration status. You'd have to call back and clarify his situation. I'm pretty sure that when my husband applied for his DL recently he showed his PR card as ID. I'm not sure if this was required or if he opted to show it for ID, but I would think that they would need you to show some proof of legal residency, and from the sounds of it, your partner is a tourist. I'm not sure you can apply for a DL in that case, but call back the DL office and double-check.