Would using Neem Oil help me to rid Tomato Plants of Leaf Miners?

It's a great product in helping control the insect from laying eggs into the foliage but will not kill the larvae already inside the leaf. Regular spraying of new growth is needed for it to work. You "must" remove and burn the infected leaves. They are territorial, after the larvae turns to a fly, it will return to the same plant and lay their eggs and start the process all over again. The larvae in fallen or picked leaves will continue to live inside and mature, this is why it's needed to burn the leaves.

I grow kaffir lime trees and sell only the foliage. They need to be perfect to sell, so my battle with leaf miners is a daily job on my farm. Neem oil is an approved organic pesticide and does work. So use it in combination of picking and burning and you'll get rid of them.

Use the internet derp

NO. Read: http://www.ipm.ucdavis.edu/PMG/r783300911.html for further information on this pest.

Yes it will, trust me.