I had a similar problem with my MBP. I don't remember, if it chimed when it came up, but the fans started up, and there was no graphics output on the screen or on an external display.
Before you go and buy a new logic board, I would recommend trying to "bake" it in the oven. If you replace it anyways, it is worth a try.
I've found this technique last week and tried it out on mine, after having the MBP stored in the closet for 2 years. It instantly booted, as if nothing happened.
1) For that, you need to completely remove the logic board from the notebook (see the iFixIt guide), and also remove any pads or plastic tapes from the logic board. Be also sure to remove the thermal paste and any dust.
2) You then have to build small cylindrical mounts (I made 5) out of aluminium foil, so the logic board won't touch the ground. You can place the mounts in the holes for the screws.
3) Heat your oven to about 220ºC (~430º Fahrenheit). When it reaches temperature, put the logic board on a tray into the oven and let it reheat for 1 minute. Reduce the temperature to 200ºC (~390º Fahrenheit) and keep it in for 5 minutes. After that, turn of the oven and partially open the lid, so the logic board cools slowly.
4) Reinsert the logic board and connect everything up again.
The reason, why this works in most cases, is, that the graphics chip is not soldered to the logic board, but mounted on a ball grid array, which sometimes causes it to loose contact. By baking it in the oven, the contacts get soldered to the chip, closing the lost connections.