How to remove dirt stains on aluminium lid of Macbook pro?

Try something like Goof-Off or Methyl Ethyl Ketone on a cotton swab on a part of the Mac that nobody see's too much to test the reaction, let it dry and look at that tiny spot a few days later.

Make sure you do this outside, both chemical have a strong order.

If nothing there, then try a little on the spot(s) and always use a clean swab, dip a little, let it work (takes some time), then remove with a dry cloth, repeat.

Don't pour liquids anywhere near the machine.

The two chemicals I've mentioned are very strong solvents, takes off a lot of things including paint.

Be warned:

There is a product called "Lime-Away" which removes calcium/lime/rust from surfaces and is marketed as "safe for most surfaces. I have seen it dissolve cast steel into goo.

Common "miracle cleaning chemicals" have directions that need to be read carefully and they should not be underestimated.

That being said, ds store has much experience in these matters and I would not disagree with his advice.