From over one to two minutes later, around a speed of 300-400 km/h, typical cycling clothes would probably be torn apart by the air resistance. But assume Lance doesn't care, or happens to wear a super-adhesive fig leaf.
Roughly four minutes after start his bike wheels would break under the centrifugal forces. This may have happened sooner or later depending on the surface, the structure of the wheels, and whether they start wobbling. Lance has lost his tires far earlier.
Less than six minutes after starting Lance reaches Mach one. Gradually air resistance would begin to heat him up. Burn marks would begin to appear starting on his forehead, hands and knees, but for philanthropic reasons we assume that Lance is indestructable. Fourty minutes later his kinetic energy exceeds that of his weight of TNT. It will become increasingly difficult to see Lance's face as he begins to be surrounded by glowing plasma.
In two hours and twelve minutes after start, having traveled 75% of earth's circumference, Lance will have trouble to stay on ground as earth's gravitational pull no longer compensates the speed at which earth curves downwards under him. Poor Lance begins to spiral into space.
Eight hours later Lance would reach the moon, in less than a week the sun, and five weeks later he would reach Pluto. Gradually relativistic effects will creep in, and Lance will see the stars in front of him change their colors towards blue, and behind him towards red. After nine years of pedaling (?), or ten years and three months on earth, Lance reaches the nearest stars. His speed has exceeded 70% of light speed, and needless to say, should he pedal through a planet, he would cause a mass extinction of whatever life forms happen to live on that planet. Even more spectacular effects - clearly observable also back on earth - would occur should Lance pedal through a neutron star.
Duh, a side note to myself: I need to stop playing with my pocket calculator.
(Thanks to an alert reader who wishes to stay anonymous for pointing out a error regarding equivalence of kinetic energy and TNT in the original post.)