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The pros of removing are a very small gain in horsepower. I've heard some members here estimate about 1-2 hp The cons of removing are that you may be required to keep all of the emission equipment intact depending on where you live and the inspection rules in your state.
Are you leaving the Cat converter on? If so, I would leave it on. Also, as stated by teamo, some states have visual and emission checks, So, if you live in a state that requires it, then this discussion is over.
Check with your state first, you may need to keep it. If you are allowed to remove it, and it still works, you may be able to sell it to someone who needs it for their state.
In addition to a very minor HP improvement, not having it there saves weight and complexity in the engine compartment (one fewer belt, many fewer vacuum hoses, the AIR tubes in the manifolds can be removed and plugged, etc).
If you have a very original car, consider keeping it, especially if you don't intend to do any other mods. Unmolested 80s are uncommon.
AIR pump takes less than 1 hp to operate and in no way causes loss of performance. If you need to pass emissions testing in your state/county, you might want to leave it in place. Otherwise, you can remove it and store it, you can remove it and sell it....or you can remove it and have an interesting door-stop.