It will dump fuel to cool the cats regardless of them being there or not. COT enabled when the car guesstimates that the cats have reached a temp higher than what is set in your tune. From the factory I think min was around 1700*f. When this occurs the car will richen up to around .75 lambda to cool the cats down and keep them from melting.
There is a 02 sensor over temp control that does the exact same thing, it will richen the a/f mix to cool off the exhaust temps. This will protect the 02's, not burn them out. If you turned these settings off and did not let the car richen the fuel mixture than you would be running the risk of burning our the 02's, and in catted sets up, melting the cats.
So to clarify, the car will run fine, just like it did when it was stock. COT will kick in when temps get to hot regardless of there being cats present or not. If it does kick on, its only protecting the car, not damaging it.
If you re curious on what it takes for COT to kick in, Im attacking a log I did when mt car on stock tune and stock exhaust. If you notice the vertical line, that is the start of my WOT pull. Where the cursor is when COT kicked on, noticed that my lambda start to drop to around .75-.76. This is bc of what I described above, the car wants to cool the exhaust temps down to protect components. It will not make your car run like crap as you cannot even feel COT kicking on. If you were to dyne the car and COT kicked on you would notice a drop in power for sure, but it's only from it running richer.
13' Mustang GT w/Track Pack: Bottle fed CJ manifold, FRPP Intake, Twin 60mm TB, Speedmaster headers, Magnaflow 3" X pipe, Bassani quad tip ABs, Jn2uned, SRP Axles, MGW shifter, BMR LCA's