It is likely a simple fix but thats all relative. The first place I would check is to get a schematic and find out what input is being triggered, likely on an IC chip when you turn on the power. If it does not turn on with the remote you can pretty much rule out the physical button on the player.
If the chip's input is going high next is to check the device that switches on the main power. likely one of those transistors on the PS board.
devices with a SMPS like your dvd player are a little more risky to work on because they do not have any isolation on the power supply because they do not use a traditional transformer.
