<p>I recently heard that the 3D printing lab at my college can do fused-deposition with ABS and PLA, but I would like to use TPU, for greater flexibility.</p>

<p>Is it possible to feed a TPU filament into the same machine built for ABS/PLA? Or is there no difference? Assume the diameters of the filaments are the same.</p>

Most printers can print most filaments. However, some more exotic filaments are not suitable for all printers:
To print a flexible filament such as TPU, the filament path from the extruder needs to be well-constrained. A bowden extruder is generally less good at handling flexible filaments than a direct extruder. Some direct extruders without a well-constrained filament path may also be unsuitable.
A printer without a heated bed won't be able to print filaments that tend to warp such as ABS.
A printer without a (heated) enclosure won't be able to print filaments that tend to warp
a lot
such as polycarbonate
A printer with a PTFE-lined hotend won't be able to print filaments that require higher temperatures. PETG is at the upper end of what can be (safely) printed with a PTFE-lined hotend, while filaments such as polycarbonate require temperatures well outside the usable range.
A printer with a standard brass nozzle won't be able to print (not long, anyways) abrasive filaments such as glow-in-the-dark.
It is likely the printer at your lab will be able to handle TPU just fine. If it is a bowden printer then you might have some difficulty printing but it might still work depending on exactly how soft the filament is and how well-constrained the filament path.