Try another user profile either by resetting the current user profile or creating an extra user account for testing. If it is identified that this is the issue, then try deleting the Temporary Internet Files or do a deeper profile troubleshooting by monitoring with Process Monitor and Process Explorer.
What other (security) software runs on the computer that could interfere?
There is no single solution to this sort of problems. Many third party programs can cause this sort of problems even after they are uninstalled, because some of them leave behind modified configuration and this is true for all browsers, not only IE.

I’ve read that some users experienced various problems with IE11 after having installed the “famous” KB3146706 Windows Update… maybe worth digging in that direction.

I have only used the following once (with Firefox and with success) so JF may want to try other suggestions first. It is called ADWcleaner. There is a description at this site:
http://www.bleepingcomputer.com/download/adwcleaner/

I was a hardware electronics tech for 38 years.
Sometimes I had to troubleshoot software to fix some machines. I found that in some cases, if restore points are made, that by going back in time to a point where the machine was working ok, then you can add your updates or re-install the third party software. Since IE is super glued to windows, you may even have to do a clean (ugh!) install of windows to cure the problem.

PS, If you do find that clean installing windows will do the trick, I hope that you have your important files backed up before doing it.

You do not mention the OS. I assume you are not using Win10, since you do not mention Edge.
I assume you have checked the ad-ons in IE. I would also assume the download location is under the User’s ID where there are readwrite permissions.

If you think it’s malware, these are my suggestions:

I would start with a bootable Rescue Disk – Bitdefender and Kaspersky have good ones but there are others out there. My experience is that you will need wired Internet – wireless drivers are difficult with the rescue disk CDs. Boot off the CDs, update the definitions and run. If you have UEFI, you will have to go through SettingsRecovery.

In lieu of a bootable malware remover, I have had success pulling the HDD, connecting it to a PC (SATA to USB) and running it against a good removal program (I use TrendMicro Internet Security).

In either of these cases, you are not loading the malware into RAM.

Next try running Malwarebytes and SuperAntiSpyware in Safe Mode without Internet connection (update before, of course, and use their most invasive scan settings). They will get rid of the bulk of folders, files and most Registry entries.

I have had most success with final cleanup using Rkill and HitmanPro in Safe Mode with networking. Both at BleepingComputer. First, rename Rkill before putting it on the computer – something like 123.com (supposedly this fools some malware that recognizes it). HitmanPro has a 30-day free trial and it will need an Internet connection to update in Safe Mode. It seems to find traces and entries that the others miss. Run rkill, then HitmanPro.

Another note. I have installed all the IE11 update patches that have come down in the last several months. I did some extensive downloading with IE11 on Win7 on Monday, so I’m pretty sure the problem is not patch-related.

If there IS any malware on board, it might be worth going to one of the free malware removal sites, where the experts there will help clean the system.

If there’s no malware, they’ll let you know all is ok.

http://www.bleepingcomputer.com/ (link is external)
http://www.geekstogo.com/forum/ (link is external)
http://www.cybertechhelp.com/forums/ (link is external)
http://forums.whatthetech.com/ (link is external)
http://qmalwareremoval.freeforums.net/ (link is external)

Hi Woody

You need to change the download folder location. Go to view downloads, click Options and change the download folder. computer may need to be restarted for this to work.

Highly doubtful that the KB3146706 update will cause IE11 to not download files. I know this because I recently downloaded a file from IE11 with the KB3146706 security update installed on Win7 SP1. So it’s most likey malware related.

Is your temporary internet files cache disk space set to 0? We are seeing a lot of these issues where I work.