Do the batteries that come with it ever wear out if you do it?
If you leave the scanner on, powered via the USB port, there is no "wear" on the batteries per se, since the scanner will not charge the batteries while it is on & scanning systems.
Rechargeable batteries can eventually 'wear out' due to too many charge cycles, or incorrect charging leading to overheated cells. But none of that is occurring as long as the scanner is powered up. You can operate the scanner via external power, even without batteries, but as mark's already noted, with no batteries, it will not record. That also results in the replay feature being unavailable. Overcharging, as well as excessive charge cycles, can result in the batteries developing a high self-discharge rate, where a battery that was 'fully charged', will run itself dead within a short time, even if it's not in use at all.
If you want to run the scanner on external power, but also have battery power available, so that the record/replay feature is available, as well as battery operation in the event of power failure, get a set of low self discharge batteries, such as Eneloops. Charge them externally, in a 'smart charger', then put them in the scanner. They'll hold a charge (without being used) for a number of months.