Is there any way to make it so that a probe will not be able to pick up power-ups that spawn on the map?

I don't think so... unless it already has something picked up. Make it pickup something your not using like a gas orb or something. Yeah, bad fix. xD Don't think theres any other way.

Wrong.
Disable the powerup and probes/drones/scv's will not be able to pick it up^^
I think that flags will crash.

Disabled Sprites, however, they can also be walked through.

Disabling it, however now it says that it is disabled.

Using the AI Junk Yard Dog does it, that one's my personal favorite.

Well I was mistaken xD

So if i run AI junkyard dog on all power-ups for neutral, then they won't be able to be picked up by a probe?

Will they still be able to participate in triggers consisting of 'move 1 power-up for neutral from location x to location y?'

yes

Problem: You can't run AIs for players above 8, so you might have to go with the disable method, or you can place it as a disabled sprite and put a Dark Templar under it.