Will Apple repair a jailbroken iPod touch (4g) that has been dropped and won't turn on?

What’s the damage? You can probably get someone locally to repair it. Apple will turn it on, and they will see that it is jailbroken. They have to turn it on to see if it works.

It was dropped and simply wouldn’t turn on. A moment ago it briefly displayed a blank, white screen, but then it shut off again.

May be a battery connection issue.

Plugged it into iTunes – it’s in recovery mode, going to attempt to restore it.

No. If it’s dropped and damaged, it’s not covered by warranty, jailbroken or not.

Attempting to restore didn’t work – I got an “error 28” message (hardware issue). I guess my brother will have to just pay for repairs or a replacement.