Hi,
If I say/write "We spoke to a person at the California Institute of Technology", this seems correct.
If I say/write "We spoke to a person at the CIT:, this does not seen correct. Rather;
"We spoke to a person at CIT." seems better.
Is there a rule about when to use "the" with an acronym?
First, please note that your example is an abbreviation, not an acronym.
For an abbreviation you simply say the names of the letters, eg for FBI, we say eff bee eye.
For an acronym, we pronounce the letters as if they form a word, eg For NATO, we say nay toe.
Now, to answer your question about using 'the' with abbreviations. It's idiomatic. There are no absolute rules.
eg We say he's a student at the Massachusetts Institute of Technology, or he's a student at MIT.
eg We say he works for the Federal Bureau of Investigation or he works for the FBI.
Best wishes, Clive