The OOM killer is more likely to kill some other process and trash your server.
Thankfully that sort of behavior has been vastly reduced since the thing was introduced, but disabling overcommit for high-reliability applications is still a reasonable course of action.
The OOM killer might eventually kill something, after it thrashes the system for a few hours.
I had a server last week in which swap hadn't been configured. A compilation job took all memory and the OOM started thrashing. Thankfully there's always one SSH session open but I couldn't kill anything, sync or shutdown; fork failed with insufficient memory.
Left if thrashing overnight and had to power-kill it next day.
Thankfully that sort of behavior has been vastly reduced since the thing was introduced, but disabling overcommit for high-reliability applications is still a reasonable course of action.