Is it a good idea to use brewed black tea into the lawn???
Are you asking if it's a good idea to brew some black tea for the purpose of putting it on the lawn, I think that sounds pretty expensive. If you're asking if it's a good idea to throw some leftover brewed black tea on the lawn, I think it's fine.
I have a lot of used black tea bag and leftover tea. Since tea is good to human beings, I wonder if it does the same to my lawn.
Tea is just dried and ground up leaves. Your lawn will appreciate the liquid more than just tossing it down the drain, but you aren't going to see any big effects. Tea is 99.99% water.
While you are at it, don't forget to sprinkle the used solid tea too. Again, not a big benefit, but a little bit of extra organic matter here and there is always a good thing.
You could compost the teabags (bags included--they're only paper). They'll compost faster if you tear them open, but if you have enough to make it worthwhile saving them, that's too many to tear open manually.
As Bill said, pouring it on your lawn will mostly benefit the lawn for the moisture you're adding.