My linguistic anthropology professor claimed that the local language was forbidden following the Norman invasion, which is why so many academic/political words in English come from Norman French. Is this true?
I don't know the context of what your professor means, but to say the language was "banned" in any legal sense is kind of misleading. In a roundabout way, the conquest is why we inherited so many administrative terms from Norman French, though.
So, before William the Conqueror becomes the Conqueror in 1066, Anglo-Saxon (aka Old English) was the administrative language used by the secular nobility and literature. The classic example of literary Old English is Beowulf, and you can see examples of administrative Old English in sources like King Cnut's laws: http://www.earlyenglishlaws.ac.uk/laws/texts/cn-1018/view/#edition,1/commentary,1_0_c_5 or Anglo-Saxon charters. The descendants of the Angles and Saxons and Jutes who came over to Britain centuries earlier spoke Old English, and the nobility was generally of the same ethnicity as the peasantry--they came from the same place, they spoke the same language.
When William the Conqueror took England in 1066, he wasn't the first foreign conqueror to do so; Swein Forkbeard had done it earlier that century, and his son Cnut was every bit as much of a Dane as his father, as far as the Anglo-Saxon nobility was concerned, but as you can see from the fact of Cnut's codes and letters to his people in England, he was more interested in ruling over England as part of his North Sea Empire, letting local lords go on lording under his dominion. Cnut's Letter 1020 to the English, for example, is a piece of propaganda written in English for the English in the interests of his hegemonic rule over the Anglo-Saxon lords. William of Normandy wasn't interested in the same thing; he cleaned house, installing his peers as the new lords in England over time, and in short brought about the end of the Anglo-Saxon nobility as a recognizable political entity (someone better with Norman history can expand or correct me here).
But it's wrong to think that Old English totally stopped after William became the Conqueror. In practical terms, yes, the nobility was nearly totally Norman, and linguistic historians agree that the secular Norman lords really couldn't be bothered with English--they didn't need to be. William of Normandy does keep up a pretense of caring about English ways for a little while, though. In late 1066-67, he issues a statement to the people of London in Old English claiming to want to keep the laws of King Edward in place, everything as it was, and so on. By 1070, he drops the pretense altogether, and we don't see further attempts on the part of the Norman overlords to bother with English.
Norman historians like Orderic Vitalis, Henry of Huntington, and William of Malmesbury insist that English literary tradition was already declining when the Normans came, and thus, William was in fact saving the day for ecclesiastical writing. No matter the case, Latin books were consistently being produced more than English ones in England, but that continued even after the conquest, where the Latin exceeded French. Still, after the conquest, we do have a handful of English books produced between 1100 and 1200, including but not limited to the Anglo-Saxon Chronicle, several homilies, and Aelfric's grammar, mostly in books that also contain Latin. English writing may have been a specialized thing for the scholars that were producing it, but it was there. The death of Old English is roughly pegged at 1154, the last entry in the Peterborough Chronicle, and at that point, the chroniclers are making grammatical mistakes and clearly don't use that same kind of Old English regularly.
And of course, while all of this is going on in the world of the powerful and/or literate, the Anglo-Saxon peasants making a living before 1066 were the same peasants making a living after it, and they kept on speaking Old English. But unlike the consequences of the Danish settlement, where a lot of Scandinavian everyday words seep into English, there wasn't a lot of permeability between English and Normans. The Normans had no interest in being bilingual, and there weren't many incentives for ambitious English folk to learn French--the kinds of people that did were in the church.
Years go by, and France and England comprise a very large and fashionable area of French-speaking nobility, so French stays strong as the language of administration. Over time, though, the Anglo-Norman French starts to look very distinct from the continental French dialects. The big break is when King John I loses Normandy, effectively dividing the nobility who formerly held lands on both sides of the channel and cutting off England from the continent. From that point onward, we start to see something like Anglo-Norman nationalism for the island for the first time, where they'd previously clearly considered France to be the lion's share of landholding.
In very broad strokes, the isolated Anglo-Norman slowly become more "English" over time. The next big linguistic landmark is the Statute of Pleading of 1362, which allowed pleas in court to be made in English as well as French. English officially enters the world of law. Now, language usually borrows words from other languages when there's a new need for a word and another language is readily available--that's why a lot of medicinal terms come from Greek, and a litany of religious terms come from Latin. Similarly here, with English now in a position to be used in legal proceedings, it just borrowed from the ready-made French words, and we get words like plaintiff, judge, domain, etc. So in general, it wasn't so much of a ban in the strict sense as a colonial breakdown of the literary tradition and no interest on the part of the colonizers to let the English represent themselves for a very long time. Hope that helps!
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