How many LEDS can be connected and ON at once to the RPI GPIO without burning out the system?
There is no documented limit. Each pin can supply up to 16mA, I would suggest limiting total current to 200mA (this is just an educated guess). The limit for older Pi (any without the 40 pin header) is total 50 mA. See Raspberry Pi Power Limitations
If you need to switch more current use MOSFETS or transistors.