I completed the over seeding (with tall Fescue) - I covered with Hay straw.

What is the procedure of removing hay straw , once grass is growing. Also when do I remove the straw?

I've never heard of using straw mulch for overseeding but maybe you mean seed placed over soil?
Assuming that is correct, most of the recommendations for cutting new grass I have seen are to cut when new grass reaches a height of 3 to 4 inches. Cutting before roots are well established (especially if you are using a riding mower) runs the risk of damaging or severing small, tender roots.
Most of my experience was in seeding large highway construction project slopes and median areas and straw mulch was never removed but simply allowed to decay. If you feel that you must remove the mulch, wait as long as possible because of the same risk as described above with respect to damaging new roots.