On a Windows machine, inspecting the memory of the IE process is limited to the owner user, and administrators. If an attacker can inspect the memory of your IE process, then that attacker has taken control of your machine and/or your account, and you are already doomed.
If the machine is "shared" then there is no really good solution: if an hostile entity can take control of the machine after you have finished using it, the same attacker could probably take control of the machine before you begin to use it, and plant a key logger, at which point your password is toast. No amount of closing/rebooting will save you. The conclusion is grim but unescapable: do NOT type passwords (or anything secret) on a shared machine.
If you are in a context where you only fear attackers who grab the machine after your usage (the "laptop theft" model), then the best you can achieve is probably to employ full-disk encryption (there are various products for that; some versions of Windows already include one called BitLocker). If you take care to shut down the machine when you are not using it, or if you mind the details about sleep/hibernation mode, then this should protect your data, including the pieces that linger in RAM.
For attackers with little competence, you could simply ensure that the IE process is dead, by closing all windows, but this would not necessarily remove all traces of the data from RAM. Your passwords may also have been written to the virtual memory file. Full-disk encryption is a more comprehensive solution.