Do I still need to remove or move my mods folder elsewhere whenever there's a patch update?

Not sure. But I always copy my saves folder into a backup folder and move my mods folder to the desktop, after update I always start the game and go into cas before I put the mods folder back, just to be safe.

I've never moved mine during any of their updates or patch, I wasn't aware that you ever were suppose to. It's not done anything negative so far at least.

I have never bothered moving mine either. My perhaps flawed reasoning is that the actual game files are being patched and not the user files. However, I start my saves over so often that it would not be that upsetting to me if I lost them. I'd just shrug and reinvent my world again. So far though, nothing has ever been messed up during any of the patches and this has applied to Sims 2 and Sims 3 as well.

Still, the accepted wisdom is to move the files out to be safe and it only takes a few seconds really.

The simgurus always say it's best not to play with mods after a patch, until the mods are confirmed updated or not in need of an update.

But you can always just try playing with mods in, and wait to see if anything appears broken before trying to find and temporarily remove the culprit.
(Make sure you make a backup copy of the save or saves you're going to play, if you do it that way.)

I didn't move my Mods folder once when I did a game repair and it totally messed my Mods folder up. I discovered I didn't have all the mods I had previously. Out of caution, I make a point to move the Mods folder to Electronic Arts, so I just have to slide it back into my game folder when things have been deemed okay.

It's always best to move the folder to a new location before patching your game. Whether or not it messes with your mods folder (or game saves) is neither here nor there, it's just simply a good habit to get into.... And it's highly recommended all the time over at Answers HQ by the troubleshooting helpers over there.

Why risk messing up any saves or your mods folder when it's such a simple thing to do?

It literally takes 2.4 seconds to cut and paste the folder to your desktop and just leave it there until you've verified/updated any and all mods/cc before plunking it back into your documents directory and turning them back on in the game options.

Whenever I start up my game and see that it wants to update/patch, I immediately close out of there, move my mods folder (and make a copy of my save file to that location too), and then start it back up to install the new update/patch..... and I only wasted less than one minute of my life to do so. One minute of my life versus losing possibly weeks/months/years worth of gameplay, save files, and downloaded mods/cc that might not be available anymore? I'll take the former because, much to my chagrin, I've already experienced the latter.