There should be no problems leaving it on for a long time unless it's sitting somewhere that's exposed to unnatural amounts of heat or magnetic fields (such as atop your monitor or something). Just make sure it's ventilated and isn't sitting too near anything that generates magnetic fields such as speakers or high power amplifiers.
Your real fear should be: How long can Windows run without crashing? It'd be a shame to get 90 percent done overnight, only to see a BSOD waiting for you when you get up in the morning.
And by the way, do you know where hyperterminal is downloading these files to, and is there enough disk space in that location?
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Tony Fabris