To say I am studying at a university:
Ich studiere an einer Universität.
I am studying at the University of Heidelberg:
Ich studiere an der Universität Heidelberg.
I am studying at the TU Munich:
Ich studiere an der Technischen Universität München.
Is the article "der" needed in these examples?
What about for foreign universities like Cambridge, Oxford, Harvard, MIT? Would they all be feminine, since they are universities? (But keep in mind that MIT is an "institute of technology".)
Yes, the definite article is definitely required in your examples. Keep in mind that technisch is an adjective that has to be inflected:
Ich studiere an der Technischen Universität München. Er geht an die Technische Universität München.
The abbreviation TU (pronounced as separate letters) is very common, too.
MIT is usually neutral (am MIT, where the letters are pronounced the English way, more or less).
Foreign universities are indeed feminine, too, though in practice you often avoid Universität Oxford or University of Oxford, referring to the place instead:
Sie hat in Oxford studiert.
This pattern is commonly used for prominent US universities as well, even when it is technically incorrect because the universities aren't named for the cities they are located in: in Harvard, in Yale.
Yes, the article is needed for the first three examples. The article are all feminine because they refer all to the word Universität.
The same rule applies to other university names if you add "Universität"
For Cambridge and Oxford it is common to omit "Universität" because it's obvious:
Ich studiere in Cambridge/Oxford.
And since MIT is an acronym for Massachusset Institute of Technology you just say
Ich studiere am MIT.
It gets the neutrum article from "Institut"
