Can any filament be used in with any 3D printer? 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.

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.
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.

The question is not what the machine is built for but how it is built. Let's break stuff down some into why some filaments work better than others and the challenges with them.
Temperature range
The first obstacle is the temperature range of the printer. For example, if your printer's Extruder can't get over 220 °C, you can't print ABS. You need to make sure your printer can match your intended filament's temperature range.
TPU usually prints around between the same temperatures for PLA and ABS, so it should work from that range.
Heated Bed
Most Filaments work better with a heated bed, but some are almost impossible to be printed without. For TPU, a heated bed should be used but is not absolutely mandatory.
Heated Chamber
Some filaments can't be printed without a heated chamber, others like ABS highly benefit from it. TPU is ambivalent on this as far as I know.
Extruder Setup
There's basically 3 extruder setups. Pellet extruders are super rare, so we don't concern about them. The other two are Direct Drive and Bowden. In a Direct Drive, the extruder motor is right over the hotend, and pushes the filament directly into it. In a Bowden setup, the hotend and the extruder motor are connected via a Bowden tube. Both have benefits and drawbacks:
A Bowden makes for a lighter carriage, leading to faster printing speeds. A Direct Drive has much less trouble with elastic filaments and can do much better with retraction.
TPU is one of the filaments that works much better with Direct Drive.
Other considerations
Some Print services do print in machines set up for one filament type and that only, because it prevents cross-contamination of the nozzles, especially when a high-temperature print material remains in it when a low-temperature print comes next. Having for example a little bit of ABS left in a printer that runs PLA next can lead to very very extreme clogging.