My USB AVRISP programmer (whose brand escapes me at the moment, but it isn’t yours) supplies +5V to Vcc by default. I think there’s a jumper on the programmer to remove this.
What should I do while programming my 3pi? Will providing +5V damage anything?
You should not use your programmer to supply 5V to the 3pi–this risks shorting two potentially different voltages together and damaging something. Also, be very careful not to try programming the 3pi while it is turned off or in situations where it might lose power (e.g. with low batteries).
Alternatively, what if I remove one of the batteries (they’re in series, correct?) and do supply power via the programmer during programming?
The 3pi wasn’t designed to be powered that way, and we haven’t characterized what the effect of doing so would be. I strongly suggest you remove the Vcc-power jumper on your programmer and just program the 3pi while it is powered from its batteries as intended.