Whats the best way to remove old peel and stick tile adhesive from a basement floor?

Heat gun or hair dryer, putty knife, and carefulness.

The glue is most likely a tar based mastic glue. It will soften with heat from a heat gun and can be scraped up. The remaining residue can be removed with mineral spirits. Be careful using a heat gun, the mastic will burn if you let it get too hot and the melted mastic can severely burn you so wear gloves. You will also need good ventilation during the process.

LESTOIL! Put some undiluted lestoil in a small spray bottle, spray it on the adhesive. Wet and wring part of a towel in hot tap water, rub it on the lestoil and voila! Your goo is gone and the vinyl isn't harmed in any way. Works wonders.

If you have a heat gun use it; If you don*t have a heat gun use a hair dryer set on the hottest setting;

It will take a lot of scraping but it will come up

Then use some white vinegar (from the dollar store, in case you need a lot) to get the rest of the small stuff up.

Good Luck

easiest way to do it is get yourself some "Level Best" made by Savogran and skim it. You've got to be pretty stupid heating up a tar indoors (which it isn't).