I recently had a large tree removed and the company also did the stump grinding. They covered the hole with the mulch from the stump. My question is what should I do next considering I am wanting to eventually seed grass in this area? Should I remove the chips or let them sit?
This is one job that I get to do quite often each year. Grass doesn't grow as well in wood chips as it does in topsoil, so be prepared for some digging. I find that a pitchfork is often easier to use than a spade shovel. Tree stumps often have a surprisingly high quantity of chips once ground out. Be prepared to take more than an hours work on just this part. The chips can be used as mulch or composted.
Once the chips are out, it's time to add topsoil. If you have a clay based soil, don't fill with sandy soil, and vice versa. The closer the soil you use the better, in terms of lawn uniformity and the soil's moisture holding capabilities.
If the hole is large, which I'm assuming is the case, fill about 12" and pack it down. You can use a tamper, tamping iron, sledge hammer (use the top of the head, not the actual hammer), or do some kind of maniac dance. Once it's pretty solid, fill another 12 inches, or to the surface level if you get there first. Tamp again. You want the surface to be slightly higher (1/2 inch or so) than the surrounding area, to allow for some natural settling, and be careful not to leave it lower in the middle, or it will cause lawn appearance issues and collect water.
Now it's time to rake the surface smooth. I use a garden/dirt rake. You want it to look natural, to blend in with the surroundings. Look at it while imagining how it would look as lawn.
The next step is the grass. Sow grass seed onto the smooth loose surface. I usually get best results using quite a bit more seed than is recommended, sometimes like 200%. Once the seed is down, lightly rake it in to the surface. I use the back of a plastic lawn rake. You don't want the seed to be buried too deeply. Just covered is ideal.
(Optional) It is best to put down moisture retaining erosion control. Put down a straw mat on a large area, or use loose straw in a thin covering. I often use the product Seed Aide with excellent results.
Keep the area watered well until the seedlings become established like the rest of the lawn.
The chips can take years to fully decompose and this process will take the nitrogen in the surrounding substrate. The more nitrogen, the faster decomposition of the chips.