I have discussed Mariology quite a bit on this blog; I have been studying its origin and development, as well as Roman Catholic and Eastern Orthodox defences of the various Marian dogmas and doctrines, for the past 11+ years now (I have been asked to write a book on the topic from an LDS perspective; I hope to do that, though it will be forthcoming for a few years, as with volumes on soteriology and Christology—I wish to write thorough volumes on these central issues, so they will be long-term projects).
One of the more popular "go-to" arguments is that of alleged Marian types in the Old Testament that have Mary as their anti-type (fulfillment). However, they often break down when examined carefully. For instance, some Catholics (e.g., John Henry Cardinal Newman; Tim Staples) claim that Mary is the New/Second Eve, appealing to Irenaeus of Lyons and Tertullian who made the Mary/Eve parallel in their writings. As Eve was created in a state of original innocence, Mary, to be better than the type, must have been free of sin all throughout her life. However, when one reads the totality of these and other authors, they accused Mary of personal sin, showing that they did not believe that Mary, to be typified by Eve, to have been sinless, one of the building blocks of the Immaculate Conception which would be proclaimed as a dogma in 1854 by Pius IX.
For a discussion of the Immaculate Conception and the theology of patristic authors such as Irenaeus, see:
For a previous discussion of another "type," the Ark of the Covenant, see:
Indeed, so flimsy is the purported typological “proofs” that some Catholics have appealed to, many have called this interpretive framework into question. Robert Sungenis, in answering a question posed by another Catholic apologist, John Salza, wrote the following. While I disagree with Sungenis on many topics (e.g., see Responses to Robert Sungenis, Not by Bread Alone (2000/2009)), I agree completely that the appeal to purported OT types of Mary to be very questionable (my fellow Latter-day Saints are not immune to this, too; some authors have twisted the Old Testament to find “proofs” of Joseph Smith. A classic example is that of Joseph Fielding McConkie, His Name Shall be Joseph: Ancient Prophecies of the Latter-day Seer [Salt Lake City: Hawkes Publishing, 1980], so I am not simply picking on our Roman Catholic friends on this point):
R. Sungenis: John, in general, I think we have to be very careful when we attempt to use analogies and allegories to prove Catholic dogma. A tendency to use proof-texting, for example, is often utilized when attempts are made to prove Catholic doctrines about Mary from the Old Testament. Some are tempted to mold the allegory so that it will fit the doctrine, and since allegories are somewhat fluid, one can usually cut and paste them until he finds an impressive connection, after which we are prompted to marvel how the Old Testament teaches Mary’s Immaculate Conception, Perpetual Virginity or Assumption. In actuality, the Old Testament doesn’t provide any factual evidence supporting these three Marian doctrines, and the New Testament can only vouch for one, perhaps two, at best. In fact, some Old Testament allegories could be fashioned in such a way to deny some Marian doctrines. Marian doctrines are supported mainly by Catholic magisterial pronouncements, and the factual evidence regarding those doctrines comes mainly from Tradition, not Scripture. (Source)