We are having the same issue here since our upgrade to 2.3.3 this weekend (just for some insight, this is a CentOS 6 box with the most recent LAMP packages). Our fix for this has been to put the moodle server in the "trusted site" list; not sure why this fixes it, but it has been a solid workaround so far. Other browsers do not appear to be affected, and only certain versions of IE are displaying this behavior as well.
File upload fails for IE9 after upgrade to 2.3.3
Just for documentation, the reason why putting our Moodle site in the "trusted sites" list worked for us is that we had the setting to force compatibility mode for intranet sites. I'm not sure if it's set by default or not, but adding the site to the trusted sites list will bypass this setting for that site only.