The oily residue is just that, residue. It's left from the thermal pads and is normal and completely harmless. Also while that doesn't look like the best application of thermal grease on the GPU die, it too won't likely be the issue that just up and kills your card. If anything the GPU would have run hotter and downclocked itself or not boosted as high. 
 
I can't really see anything in the PCB shot you uploaded since it's a very low resolution pic so I can't spot any blown caps or any other issues with SMDs. Though if you've tried it in another PC with the same results, the card is likely dead. 
 
You could try stripping the card down completely and baking the card to possibly reflow any cracked solder joints and "maybe" resurrect the card. Though be warned, it's generally not a good idea to do this in an oven you cook food in. But having said that you'll want to elevate the card off the rack by using something like a few pieces of ceramic tile, lay the card flat on the pan and bake it at 400F for about 6-8 minutes. Take it out and let it cool down then reassemble and try it in your PC again. 

It's with my deepest sorrow and misery of finding a replacement that I inform you of the death of my EVGA 980 Ti. 
 
So, I baked my card not thinking the problem was anything related to cracked solder. But lo and behold, after doing so, reassembling everything, and installing into my PC it booted! But my joy was short-lived as not soon after a plume of smoke and the rancid smell of burning plastic sealed the fate my beloved 980 Ti. Upon immediately cutting the power to my system merely seconds after catastrophe, I ripped the card from the bellows of my PC and took note of the damage. I'm not sure what component on the card is torched, but I'm not optimistic it's anything that can be fixed. I'm not sure what went wrong, maybe during the bake solder bridged a connection causing a short? Either way, gaming is on hold for the unseen future as affordability of a new GPU is well beyond my current financial position. Two kids, mortgage, a slew of bills, and repayment of student loans takes priority as the sole provider to my family. At least its winter up here in the north so the depression can be dismissed as seasonal. But cryto miners and scalpers are relentless at keeping the cost of an affordable GPU within the reach of your average born in 1985 just trying to get by and live a modest life while raising two young kids and barely scraping by generation.
 
Please say some kinds words before I remit my 980 ti to its eternal slumber, deep beneath the granite laced dirt surrounding the foundation of my domicile.