I haven't heard of any recent breakthroughs. The strongest result that I know is due to Misha Bialy:
Theorem. If almost every phase point of the billiard ball map in a strictly convex billiard table belongs to an invariant circle, then the billiard table is a disc.
Stronger results are available for an outer version of the Birkhoff conjecture. Tabachnikov
proved that if the outer billiard map around a plane oval is algebraically integrable then the oval is an ellipse (linkarticle,arXivversion).