Shadal says that of course we find Morenu (מורנו) all the time, but the question arises from older manuscripts where there is something which gives rise to his question. For example, in Rabbi Yehuda Ha-levi's Diwan (a rare manuscript which he purchased and was about to publish) we always find מרנו (lacking a vav). Sometimes we even find it abbreviated and including nekkudot, like this: ' מֳרֵ. He points out that very old manuscripts sometimes have older readings, so such things are not mistakes at all.
In fact he concludes that this is the primary reading, and we see from this that the term מורנו is really מֳרנו and derives from the word denoting lordship rather than teacher. The term actually comes from the Aramaic phrase מרנא ורבנא, where the absolute form of the word is מָר, master, not מוֹרֶה, teacher. The "German-Polish" (Ashkenazic) pronunciation of the kametz with each failure to differentiate between the two kinds of kametz lead to the pronunciation as Morenu, rather than Marenu. From this it was but a small step to spell it maleh with a vav, מורנו, and this implied that the word means to say "Our Teacher." This is essentially what happened with the word haftarah (הפטרה) which is often pronounced and written haftorah (הפטורה), but there is no doubt whatsoever that the correct reading is the former.
Shadal included this, slightly expanded, as a footnote to his edition of the Diwan (link), which I will get to soon.
What he did not mention (or know) is that there is actually a whole spiel about the correct pointing of the word מר, which undoubtedly means 'master.' Although we Ashkenazim pronounce it "Mar" the Sephardim pronounce it "Mor," with a holem. The first explicit reference to this I could find says it's a mistake (naturally). In R. Elijah Levita's Tishby we find the following entry:
He says that there are those who write it מֹר, but this must be a mistake, because if this were the correct reading why not [ever] write it with a vav?
The answer to his objection is that they definitely do write it with a vav, he just never saw it.
Although many people will want to point to the writings of the ge'onim (as I will) you can actually find it numerous times in the Mishnah commentary of R. Ovadiah Bertinoro. Here is one example from the first printed edition (Venice 1548):
If you don't find it in your copy - it is spelled מר rather than מור in the Bar Ilan database - it's because later editions corrected it, but this is how Bertinoro always spelled it (or at least how it is spelled in the earliest printed editions).
We can find it spelled מור in numerous manuscripts from the time of the ge'onim. Here are but two examples:
The first is a description of a manuscript in Ohel Dawid, the catalog of the Sassoon collection. The second, below, is from Halakhot Pesukot. There are many more such examples, including the Sefardic rescension of Iggereth Rav Sherira Ga'on.
Thus we see that Shadal did not seem to realize that pointing with a holem (Mor) is undoubtedly possible. Practically speaking I'm nor sure what difference that makes, since he'd be right in any case that it meant "master" and not "teacher." Actually the Sefardic pointing supports him.
Interestingly one can find a clear support for Shadal in Rashi on Kiddushin 31b which discusses the proper way a person should refer to his father. Rashi ד"ה ואמוריה explains אבא מרי to mean אביו ואדוניו. Not only that, as it happens in our texts of Rashi מרי is written with a vav, מורי. I will leave it to the text experts to get to the bottom of it and determine if this is the correct reading of Rashi. Incidentally, Soncino translations מרי as "teacher" and Artscroll translates it as "master." Also see Rashi Bava Kamma 49b where he gives the exact same translation for מורי (this time the Gemara reads it with a vav).
Interstingly enough in the 2005 edition of the Tishbi there is a note in this entry by Rabbi Meir Mazuz which says that he heard from his father R. Mazliach that the Ashkenazic reading "Mar" is correct, but the Sefardim changed it to "Mor" because "Mar" connotes bitterness. And, I would add, "Mor" means "myrrh." Actually, he brings a quotation from the Tikkunei Zohar which makes a play on "Mor" as both "master" and "myrrh," which implies that the Zohar (*coughSpaincough*) read "Mor." (Link; this wordplay is very sbutle, but I agree with him. It is a nice catch. Almost as if sensing that the wordplay doesn't really work with Ashkenazic pointing, the editor of this 1909 edition with nekkudot doesn't point "Mar," seemingly the only unpointed word on the page! That's pretty cool, except that he was Sefardi. צ"ע.) I also want to point out that this edition (Machon Harav Mazliach 2005) is the one to get, if you ever wanted a Tishbi. The only criticism I have is that for some reason they changed R. Elijah's unique vowelization of many of the entries, a major major error on their part. I can't begin to fathom why they did that, and if they had to do it they should have noted each change in the notes. Apart for that, it is simply the edition to own with its manifold notes and supplements.
Here's Shadal's comment in his Betulat Bat Yehudah (Vienna 1840) pg. 111:
I would just note that the typesetter spelled the book Besulath Bath Jehuda, as you can see. In other words, in the same word (בתולת) he renders a תי"ו רפויה as /s/ and also /th/. That's enough to drive a pedant crazy.
In this footnote he expands his thought, mentioned only briefly in German, that we have a great source in old manuscripts to teach us how the ancients read, and from this we can learn many things. He gives another example which I admit that I find puzzling. The example is גאוון, which he says is a faulty modern spelling, for in old spellings it is always גאון with one vav. He says that the old readings prove that the traditional spelling with one vav is correct. I think I'd have a better idea of what he was getting at if I could find many examples of "גאוון" in print but I cannot. So I'm not sure which moderns he has in mind. I guess unwittingly his own writing became an example of what he is talking about. We now see that in 1840 people were spelling it "גאוון," evidently a passing fad.