Someone here may be able to help you, but I would suggest asking over at r/academicbiblical as well.

If what you said is an accurate representation of what he thinks, your teacher is, to borrow a trusty phrase of Jerome, stercora confecit.

Point by point:

New Testament was written by the Romans

Not even a little bit. The closest you come is that Acts claims Paul is a Roman citizen, something which is never claimed in Paul's letters.

relieve themselves of guilt for the death of Christ, and place it all on the Jews

In the period when the four canonical gospels and the pauline letters were written (some before 60 CE, most before 80, all probably before 100, by 150 at the latest), there is no clear distinction between Christians and Jews, either to outsiders or within the communities. Such a claim therefore makes no sense, and has no basis in historical reality.

New Testament also attempts to strip Jesus of any sign of Judaism

You might suggest that he actually read the New Testament, specifically the gospel of Mark, because there is no way he could have actually done so and still asserted this conclusion.

EDIT: So in case your teacher actually is a raving loon, in order to get something out of this class, make sure you at least read these two books:

• Nirenberg, David. Communities of Violence: Persecution of Minorities in the Middle Ages. Princeton: Princeton University Press, 1996.

• Browning, Christopher. Reserve Police Battalion 101 and the Final Solution in Poland. New York: Harper Perennial, 1998.

The way you can tell if he's a raving loon is if he has you read Daniel Goldhagen's Hitler's Willing Executioners and seems to think it's right.

I have heard this hypothesized about the narrative where the Jews say "his blood on us and our children", removing any sort of responsibility from Pilate. The reasons for that suspicion include the historical issues with it. The description of the Sanhedrin doesn't match what we know of it from Jewish texts, Pilate being reluctant to execute someone doesn't match with Josephus' description of his brutality, and his releasing of a prisoner on Passover (a supposed Jewish custom, as far as I am aware attested nowhere else) matches with neither his brutality nor his lack of respect for Jewish custom, which caused a great deal of unrest in Judea during the period. Wikipedia says that the Anchor Bible discusses this, which I unfortunately don't have access to.

However, that's far too broad a claim.

my teacher said that the New Testament was written by the Romams in an attempt to relieve themselves of guilt for the death of Christ

Well, it's possible that the authors were Roman citizens. Likely, even--Paul seems to have been one. But ascribing the motive of removing guilt for the writing of the entire NT seems farfetched, given that it's a small part of the NT. And while the narratives share Pilate as absolving himself of the execution, not all of them blame the Jews as a group--Mark, for instance, seems to blame the priests and a mob of people in Jerusalem.

He went on to say that the New Testament also attempts to strip Jesus of any sign of Judaism.

This isn't really right. I mean, Jesus is usually portrayed in contrast with the "Jewish Jews"--the priests and Rabbis. But the authors of the NT take great pains to identify him with Hebrew scriptural passages and prophesies (even when it doesn't make much sense). And they use Hebrew/Aramaic terms, like Jesus calling God "Abba" (Aramaic for dad), or his followers addressing him as "Rabbi". It leaves a pretty significant non-Roman element to the narratives.

What you write would be controversial at best and just plain wrong at worst, it's hard to distill exactly what your teacher said.

But there is a hint of truth in it which makes it a worthwhile topic to discuss. There's two important facts about early Christianity which relate to this. Firstly that the Jewish people in Jesus' time were one of the most rebellious peoples in the Roman Empire. The relationship between the various Jewish states and the Empire were a cause of anger for many Jewish groups. And therefore loyal Romans were quite concerned with Jewish insurrections.

The second part is that Christianity blossomed on the intersection between Judaism, Hellenism and Roman citizenship. In part this seems to have already happened during Jesus' lifetime, but after his death this really became its fertile ground, more than people who clearly and unambiguously self-identified as Jews. The authors of the New Testament wanted to make clear that Christianity was not linked to Jewish insurrection, because that would make it less appealing to people who were closer linked to Hellenistic and Roman identities. Also of course that would increase the suspicions on the group by Roman authorities.

This is not to say that Jesus was advocating Jewish independence. Just because the text goes to considerable length to explain that he didn't, doesn't mean that he did. But it does mean that some people thought that he did, and that the writers were afraid Jesus would be seen as a Jewish fanatic. Gingerkid already made an important quote about this, the blood on their heads, but also Jesus' evasive answers about kingship for example. Although there is another explanation for that as well: if Jesus was the messianic king, how come he got crucified? The NT transforms a figure that is actually mostly political and ethnical the OT, the messiah, into a purely spiritual and universal figure.

In summary your teacher is wrong, but the point he brings up is indeed a painful one.