House fan installed in skylight - good or bad?

[Verthein]

I have a large old rowhome in Baltimore with a deteriorated cupola-style skylight above a shaft that extends all the way from the flat roof down three floors and into the basement. I assume it was originally used as a chimney to get cooler air from the basement up to upper floors, and evacuate hot air. The top two floors access the shaft with windows. The walls between the shaft and the attic crawlspace are anything but airtight. My contractor suggested rehabilitation for the skylight and adding a fan to one side of the skylight (vice a window), thinking that we could switch the fan on for 15 minutes when we get get home at the end of the day and remove hot air from the top floors (and attic) before lowering the AC temperature setting. I was thinking that the only hot months of the year where we would want to do this are also very humid here - yes, the contractor's plan would mean less loading on the AC units, but it would mean drawing in humid oputside air whenever the fan runs, which would have to be removed with an even lower AC temperature setting or by buying dehumidifiers. I'm wondering whether this skylight fan would improve energy consumption and comfort, or whether I'd be better off with a sealed skylight shaft, and just leaving the AC turned on June-August?

[mark beiser]

Without assessing the house in person, nobody can tell you for sure if that is a good or a bad idea, but chances are that it is not a good idea, possibly even an extremely bad idea.

The very existence of an open shaft running from the basement to the roof is likely a bad thing, especially if you have combustion appliances in the basement.

[Verthein]

The very existence of an open shaft running from the basement to the roof is likely a bad thing, especially if you have combustion appliances in the basement.

Bad idea because...? No fire break?
They used to put kitchens in the basements of the big houses back then. I have two chimneys that start down there, too.