I've tea dyed fabrics after pre-washing, before making the quilt. I've also tea dyed quite a few of my clothes to soften their colours. I love it, drinking-strength tea creates a lovely gentle antiqued look on white and cream, and it can take the "edge" off garish colors as well.
I have noticed that (using actual tea bags, not a commercial product) it comes out a bit different every time. So I'd be a bit wary of tea dying a finished quilt... :)
What I do:
I make sure I wash my fabric first to get all the chemicals out, with my normal detergent in the washing machine.
Then I make tea. For antiqueing fabric, a large pot of tea with anything between three and six tea bags, depending on how much darkening I want. (For darker dyeing, I'd use a stronger brew and longer soaking time.)
Any black tea will do. Redbush (roiboos) tea works, too, and gives a lovely, slightly more reddish beige/brown than black.
I leave the bags in for maybe ten minutes, again depending on how dark I want it, then pour the tea into a bucket two thirds full of cool water, leaving the teabags behind in the pot. This gives me a weak tea-ish liquid, I can see the bottom of the bucket through it!
I dunk my pre-washed fabric into it and swish it around a bit, then leave it for maybe two minutes, maybe ten, depending whether I want a slight "yellowing" or a stronger brownish tinge. When it looks good to me, I take it out, rinse it, wring it out. I might put it back for a bit more dyeing if it comes out too pale.
When done, I tend to wash the fabric in the washing machine again before I use it, as normal with detergent, just to be doubly safe that no tea remains that might later stain adjoining fabrics. (I doubt that this is necessary, I'm probably being paranoid...)
Hope this helps. Have fun and good luck!