I hear of teams doing this a lot WITH screws still attached and I also hear of teams running motors doing this WITHOUT screws attached. Personally, I’ve done the first during a desperate time in a tournament. From people with experience of doing the second scenario, how well does it work? Is it recommended?
Definitely recommend doing the latter. Removing the screws and just using a tight zip-tie has never failed for us, and also helps to check internal gearing. Plus, sometimes it helps to just remove the casing to get to hard to reach spaces.
Putting Zip-Ties on your motors is a very effective and fast way to replace motors if the motors were to break. They can seriously reduce the amount of time it takes to replace broken internal gears as well. I believe I was told last year that removing the screws on the motors and replacing them with zip-ties slightly improves the efficiency of the motors as well(I was told that by my mentor at my last school, but finding the source of the study for it again will be hard). So far I have not seen a downside to having Zip-Ties on your motors other than more space being taken. In my personal opinion, if you have space to have Zip-Ties on your motors, I would highly recommend taking the screws off of your motors and Zip-tie the motor together.
Hopefully this helps 
- [TVA]Connor
Somewhat related question - what size zip ties do most teams use?
I wrote an article about zip-tie-ing motors: https://renegaderobotics.org/zip-tie-motors/
We use the large-size ones that VEX sells.
Robosource is much cheaper than VEX.
Sorry, I didn’t mean to imply that one had to get them from VEX (we don’t; we get them at the hardware store). I was just referencing the ones VEX sells for its particular size.
Oh, I wasn’t 100% sure I interpreted what you said correctly and I guess I didn’t. But I figured it was good to spread the word of Robosource to those who don’t know.
We use two 4" zip ties. This allows us to get it very tight.
On a related note, one of our teams recently attached two of the motors to their drive with zip ties. I thought it was a crazy idea, but it actually worked really well. The axles were below the C Channel making conventional attachment a bit more difficult.
OMG! At first when I heard about this, I thought people meant ziptie-ing the motor to the FRAME of the robot! Haha! Now it makes sense - great idea because replacing internal gears must take like 30 seconds. 
My teams zip tie all of their motors on, and usually remove the casing screws as well. Never had a problem, and often improved performance. Discussion with several engineers lead us to believe that the reason we get improved performance is from the common over-tightening of the casing screws by students changing out the gearing. There is a optimum torque ratio for those screws, as designed, that students are likely to surpass. Using just zip ties to hold the casing together is likely to not have the casing on too tight. We eliminated the mounting screws because why not let the zip tie do double duty, makes repairs faster and easier. If you do that to all 12 motors, that’s 120 less screws on your robot.