Say I drill a big hole through Jupiter and shoot a space probe through it. That would speed up the probe like a sling-shot maneuver, right? So neutrinos passing through a planet should also increase in energy, every so slightly, right?
I suspect if I’m wrong I’m also not bright enough to catch it, so I haven’t done the math. 
No and no. Slingshots help add speed in the frame of reference of the Sun, with the extra energy coming from the Jupiter’s orbital speed. Also, the spacecraft needs to change direction as it passes Jupiter, heading closer along Jupiter’s orbit than it was when it approached. If you had a hole through the center of Jupiter, and shot your spacecraft straight through the hole, you wouldn’t get a speed up.
There’s also a benefit from using thrust closer to Jupiter, but your neutrinos won’t have any rockets on them, so that’s not relevant here.
Sheesh. I seem to have ignored some basic Newtonian physics and then missapplied the resulting wrong physics, since my thoughts from earlier now make no sense in either framework.
But you when you refer to the Sun’s frame of reference, is that just because it’s the reference frame that actually matters, and the speed increases in all reference frames except those that experience the exact same accelleration as Jupiter or the probe?
In the reference frame of the planet, things speed up when they’re approaching the planet, and then slow down again when they’re moving away from it, and the changes in speed balance out. But the planet is itself moving relative to other reference frames (including but not limited to that of the Sun), and the object that moves towards and away from the planet will in general change direction, so it’s possible to get a change of speed that does not balance out in some other reference frame.
In principle, this applies to everything, including neutrinos. But in practice, because neutrinos already travel so rapidly, the change would be incredibly small for them-- You’d be talking something like parts per trillion.
Couldn’t have said it better myself. And I didn’t. 