OK, I need you to clarify things a bit. Youre saying that if you leave your computer idle for any period of time IE will no longer work when you come back.

What do you mean it doesnt work? does it just not open, or does it just not connect to the internet (i.e. give you a page could not be found error) but if you reboot everything works fine, unless you let it go idle? does this only happen on one of the computers?

and also have you scanned BOTH computers for virusis/adware/spyware? if they are networked then when one has one the chances of it being passed back and forth is very good. take both off the network and scan in safemode witha up to date scanner.

is the computer pulling a valid IP address from the router? are you using any type of filtering ont he router (wep keys, mac filters, limited dhcp addresses, specified static IPs)

is the router configured correctly? are you using the routers software to connect to the wireless network or are you using the software that comes with windows? Make sure both are not running, this can cause a confict which can result in you being kicked offline.