I have to write a literary criticism essay and I am unsure if it would be innappropriate to abbreviate/acronym the titles?The titles are A Farewell To Arms and A Clean, Well-Lighted Place
Yes, it is appropriate to abbreviate titles, especially of works that your refer to several times during the course of the essay.
Based on the guidelines of the Modern Language Association, your first reference should state the title in full, and then in subsequent references, you can use a shortened title--in your case, Farewell for A Farewell to Arms and Place for A Clean, Well-Lighted Place. If you are using a Works Cited page in which you list the two books, that would be the place for the full bibliographical information, and you could use the short title even for your first reference in the text. If you do not use a Works Cited page, then you would use the full title when you first cite the text and, for any references after the first, the short title.