What ultimately happens to a landfill?

I can't speak for everywhere, but most dumps in Canada eventually become parks or reclaimed land. Gases from rotting garbage is siphoned off and in many cases used to create electricity, to power the park ie washrooms, lighting and in some cases the park equipment such as lawn mowers, snow blowers.
Every landfill has a lifespan. This is based on the containment field, size and so on... Not sure what Fresh kills are...
Many cities are now creating less 'garbage' and are collecting compost-able materials, recyclable materials and so on. In my city, they have put some serious goals into place and within the next 10 years, 90% of garbage will not be put into landfills.
We are already recycling upwards of 50% of paper, metal, glass and plastics. Electronics are being collected separately and while it was once common place to send this material to Asia, this is now frowned upon, and I do believe illegal in many provinces and states. Compostable stuff is being made into compost and resold at plant stores as mulch and so on... and these markets will only continue to grow.

The real goal of landfill is "Out of sight, out of mind"

Fresh Kills:
http://en.wikipedia.org/wiki/Fresh_Kills_Landfill
also, you bring up another question. What is the process of electronics Today? Right, we used to ship off the toxic materials to African and Asian countries (often under the guise of "donations"). So, supposing that doesn't happen any more (unlikely), are there subsidies or other incentives to encourage companies to go through with the costly process of separating and recycling electronics? How does that process work?

In America the ultimate plan is to seal the trash in an air and water tight environment, monitor the water under it to make sure that it is not leaking, and then use the area over the dump for something. Because no air can get in decomposing stops, and is not intended to resume for about 1000 years, when the lining starts to break down.

Or so I read someplace years ago. I might be wrong....

future generations dig them up and wonder who the hell we were

What happens to land fill?

I have this idea and concept for a scifi story that is set in the future,
when the next species has populated the earth and that what they dig up and use as their "natural" resource, (because there are no truly natural resources left), is really just all the undecomposable non bio-degradable plastics and man made fibres we have dumped in land fill.
They have drilling and digging platforms, a whole economy, to free the precious man made fibres so they can use them.
Our rubbish is their black gold.

But seriously if the human race survives past the total consumption of all natural resources and "fuels" most land fills will become gold mines.

All the best,