Would the LEDs burn?
Maybe. It's certainly not safe.
LEDs always need at least a resistor, high power LEDs need current regulation.
AC-DC adapter (output 9v) directly connected with 3 LEDs in series.
Would the adapter become too hot and cause fire something like that?
Depends on whether or not it's a regulated power supply. If it's for powering battery-powered toys then most likely it's not regulated and will output a lot less then 9V when you put a load on it.
The way to tell is to measure the voltage with no load. If you see much more than 9V then it's not regulated.