How are people dealing with updates that require/force IE (or another app) to close?
Most organizations don't patch desktop applications during the workday, so this is not normally an issue.
but a lot of people shut down their computers at night so that's not ideal.
This is a human resource and operations management issue. If Technology Management requires that systems be powered on once a month to install patches, that's a behavior that will have to be trained. Implementing non-optimal patch management procedures because business practices are not optimal for the patch management to occur is ignoring the core issue.
The need of the organization is to apply patches to desktop systems in a timely manner while minimizing interruption to the user is the core issue. The user must cooperate in that venture. If the user fails to leave their system powered on when instructed to do so; then by all means do what is necessary (including disrupting that user's workday) to get the patches installed. I promise you.... the user will eventually learn the preferred behavior. ;-)
But don't *plan* to disrupt all users' workday without first trying the better solution. :-)