The 19-year-old McIlroy, a native of Northern Ireland, was the first-round leader by three strokes, but his even-par 71 in the second round included three costly bogeys.
In shooting his 67, Australian Rumford, the defending champion, overcame his only bogey at the par-4 fifth with five birdies overall, two in the final four holes.
"It's always difficult to follow up a really low round," McIlroy said about his 8-under 63 on Thursday. "I've learned that before. My iron play was not as good as yesterday (but) I holed a lot of par putts on the back nine to keep me going, so a 71 was pretty good."
Argentina's Juan Abbate (67) shares third place at 7-under 135 with Spain's Alejandro Canizares (68) and Switzerland's Jean-Francois Locquin (67) and four others share sixth place at 6-under 136.