No, there's nothing turned on by default in Windows that will automatically delete temp files. Not on shutdown, not on reboot, not ever.
The Disk Cleanup tool has an option to do that when its run.
The proper way to handle this is for your web application to clean up after itself in some way. You might have it delete orphaned files when a user logs out, for example. Or a (small) random chance that a "cleanup sweep" will be made on every request. Or a scheduled task that must be installed and configured to do cleanup. Or a scheduled task to run Disk Cleanup.
Do not go about just "leaving stuff" in the temp folder in the hopes that something will come along and clean up your mess for you.
Also, when you do go about cleaning up your files, only touch the files that you've created and no others. Other applications may need those files. If tracking your own files is too much trouble, create a folder that you can safely empty at will.