Hi Dariusz,
Almost all projectors I know of the last at least 15 years have a UART communication protocol (by OSRAM) between the projector main board and the lamp ballast circuit.
You cannot start the projector unless the communication between the ballast and the projector is established correctly.
In brief, a working lamp with the right resistance and power characteristics needs to be connected to the ballast circuit, otherwise the projector goes off after a minute or so.
So it's not an easy task to replace the bulb by a LED array. You need to decode first the UART communication under normal operating conditions and then emulate the ballast control circuit by a microcontroller that reads the messages from the projector main board and sends back the correct signals to fake the ballast.
I have tried to work this around with my Acer P1510 projector but not finished yet (you need a logic analyzer to analyze the UART signals during normal operation and then a microcontroller like an arduino to fake the signals by writing the right program).
Hope that helps a bit.