I completed 'Ring-a-Ding-Ding!' and got the chip from Benny, but I'm unsure whether to take it to Mister House or Yes Man. I want to support the NCR, but they have no use for the chip. Mr. House seems like he'd know what to do with it best, but I'm not sure if helping him would affect my ending with the NCR.

Upgrading the bots does nothing to affect NCR, but if you do too many missions for Mr. House, you can't do anymore missions for the NCR

Bot is correct. If I recall correctly, THAW P IIII is where the warning quest 'Don't Tread on the Bear' pulls up. Mr. House is one of the 'Independant Vegas' factions, Yes Man being the other.

If you're not going with the NCR, there's no need to give the chip to either.