Rose / Bad Wolf used the powers she got from the TARDIS to resurrect him, and his resurrection stuck, but there's no evidence that he retained any connection to TARDIS energy after that. We don't see any of that energy entering him, like the Mire device entering Ashildr to make her immortal in Series 9. The energy only enters Rose, and she uses the power from it to change reality, destroying the Daleks and resurrecting Jack. She isn't pouring out the TARDIS energy directly like we saw it pouring into her.
ROSE [to the Daleks]: You are tiny. I can see the whole of time and space. Every single atom of your existence, and I divide them.
She's controlling reality to destroy the Daleks, not pouring TARDIS energy into them to kill them. Same thing with Jack.
In the Series 3 finale, when the Master takes the TARDIS and cannibalises it into a paradox machine, there's no sign that Jack feels any effects from that. He is the one who finds the TARDIS, but after getting past the Toclafane he boldly fires a sub-machine gun into it without flinching.