I can't download the picture, it's too big for my slow internet connection but from what you describe, it uses the BTA16 in series with the power line to act as a switch and turn the lights on. The BTA16 600B is a 16A triac, not a transistor. For some reason one of them is conducting all the time, the most likely cause is the BTA16 600B has gone short circuit but you may also have an interference supressor connected across it which has shorted as well. I'm guessing the four white ICs are opto-couplers and while it isn't impossible that something before them (to the left of them in the picture) has gone wrong, the chances are the fault is on the AC side (right) of the board.
If you have a test meter, while disconnected from the mains, measure the resistance across the first and middle pins (with the component upright and the part number facing you), it should be very high value, at least 100K. If it's lower, try replacing the BTA16 600B. Make sure you get one with the same part number because the BTA16 is a family name and is made with different numbers and end letters.
Brian.