Fisher, who began the month by winning the 3 Irish Open, went to the turn in his Johnnie Walker Championship at Gleneagles second round in just 30 strokes.
But he trailed playing partner Molinari by one after the Italian had four birdies himself on the back nine.
Fisher's debut at The Celtic Manor Resort was clinched last week by results at the Czech Open - he was not playing there - while Molinari found out after his opening 68 in Perthshire that Ross McGowan's shoulder injury had sealed his first cap.
During the 3 Irish Open Fisher had a chance of achieving the European Tour's first-ever 59 and he was back in that mood with birdies at the 11th and 12th and then four more in a row from the 14th.
It nearly became five, but his ten foot putt for an outward 29 lipped out on the long 18th.
Molinari, whose brother and Omega Mission Hills World Cup-winning partner Edoardo is in the mix for a wild card on Sunday, led by one from not only Fisher, but also Scot Stephen Gallacher, Frenchman Julien Guerrier and overnight leader Richard Finch, who was among the later starters.
Play had to be suspended for 30 minutes by fog at 8.15am.
The four players fighting for the last two automatic spots in Colin Montgomerie's side - Peter Hanson, Miguel Angel Jiménez, Simon Dyson and Alvaro Quiros - tee off this afternoon.
Fisher started the front nine with two more birdies - he left an eagle chance short on the long second - and that made him eight under for the round.
At Killarney, a par-71 course, he had needed two birdies in the last four for a 59, but parred them for a 61.
Here, with the par for the course one more, he still required five more in the last seven.Molinari had bogeyed the first, but Boyd, who lost a play-off at the Czech Open on Sunday, eagled the 16th and birdied the 17th to leave him and Fisher level at the top on eight under.
Fisher was unable to continue as he had started. A drive into sand on the third led to a bogey and by failing to get up and down from just off the fourth green he dropped another shot.
It left European Tour rookie Boyd out in front on his own at eight under with eight to play, and Molinari's birdie on the second put him joint second with Guerrier, while Fisher was now part of the group two behind.