If you have a very nice lawn, then you are going to be rather disappointed
if you do not remove the stump and all of the main roots that branch out
below the lawn.

Mushrooms are the scavengers of the plant world. They cannot grow
unless there is dead and decaying material around for them to scavenge
for growth and energy materials. If you allow the stump and roots to rot
in your lawn, you will get to observe this process first hand on a large scale
beginning 1-3 years after you cut down the tree and lasting for a decade
or more after the mushrooms start to appear.

Your lawnmower will not cut low enough to remove them. And some of
the mushroom varieties are as hard as wood and they leave a large divot
in the lawn when you pull them out.

It is common to notice a circle of mushrooms in some guys lawn, along
with several lines of mushrooms branching out radially from that circle.
These mushrooms define the location of a former tree and its main branches
and they will continue to do so for many years.

If you are concerned about your lawn, then:
1) Grind the stump down as far as reasonable, but at least 8".
2) "Chase" the main roots with the grinder and grind them out completely,
don't just shave off the top of them.
3) Remove as much of the chipped wood as possible from the dug up areas.
Raking is a minimal step, but screening the soil/chip mix is better.
Very
fine bits of wood are ok, but significant chips will feed mushrooms.
4) Reseed the areas.

You need to consider how fussy you are about your lawn and how much effort
and money you put into it each year. Then consider the cost of having the
stump and roots removed versus the hassle of mushrooms for a decade or so.

I spent hundreds removing the stump and chasing the roots of a 70' maple
that I had removed from my front yard several years ago. I'll do the same
in the next year or so for the second 70' maple which I now want to remove.
I consider it money well spent. Some of my neighbors think that is wasteful
and they are content to live with their mushrooms or remove them weekly.
None of us are "right or wrong", its just that we view the situation
differently
and have different priorities.

If you want to conserve money but still get rid of the stump and some of
the roots very close to it, then I'd recommend the saltpeter (KNO3 - potassium
nitrate) plus kerosene approach. Properly done, this is quite safe, especially
these days with the deflagration (explosion) inhibitors that are blended with
the saltpeter in commercially available "stump remover" chemicals bought
at garden centers. There is a lot of information about this method available
on the Internet. I believe that a few months of soaking are recommended
to permit the KNO3 to soak down into the roots. I also believe that the fire
should smolder for days as it works its way deep into the root zone.

Cheaper sources of KNO3 are around. Some fertilizers such as "K-Power"
(Vicksburg Chemical Company) are basically pure saltpeter. The chemical
analysis on the bag should be about 13-0-44, which is the ratios for pure
KNO3. I'm moderately certain that these fertilizers also have the
deflagration inhibitors in them.

Just be warned that if you make a lot of inquiries into purchasing KNO3,
you run a modest risk on getting on a terror watchlist. If you mail order
the KNO3, you will certainly get on a terror watchlist. If you make inquiries
about KNO3 on certain newgroups, you will definitely get on the list.

Final caveat - follow the directions on the stump remover packages. Also
do a bit of Google searching for additional comments on using KNO3 plus
kerosene/diesel fuel. Personally, I've advised a number of people on this
method but I've never used it myself.

Good luck,
Gideon