In Buzz Aldrin's book Encounter with Tiber, he mentions that some of the Space Shuttle's fuel tanks actually made it to orbit. Did this actually happen, and if so, for how long did they stay in orbit, and what altitude did they achieve?
Based on this description of the Space Shuttle flight profile, no external tank would ever have completed so much as a single orbit. An external tank would achieve essentially the same orbital apogee as the orbiter itself, but that is all. The shuttle fired its OMS engines to achieve an actual orbit AFTER tank separation. This means that the tank remained on a trajectory which would intersect (re-enter) the atmosphere with no chance of ever completing an orbit.
The only way an external tank could remain in orbit would be if it was left attached until after the orbiter had completed its OMS burns to establish an orbital trajectory. To do so would increase the mass to which the OMS engines must apply the required delta-V... by more than the vehicle's payload capacity. This would require reducing, or more likely eliminating the internal payload. The upshot being that a shuttle flight in which a tank is inserted into orbit (if it were even possible) would be a mission to insert a tank into orbit and almost certainly accomplish nothing else. See this question for a more comprehensive analysis.
Aldrin has apparently exercised a bit of creative license; the title you mention is after all, a work of fiction.
There are generally speaking 2 conops to get into orbit with the Space Shuttle. The first one, and most commonly used, requires two thrusts post Main Engine CutOff (MECO) to achieve orbit. This one drops the ET in the Indian Ocean. The "Direct Insert" requires only one OMS maneuver, and the tank landed in the Pacific Ocean. So the External tank might technically have been in orbit briefly, but in a reentry orbit.