What is the term for this ambiguous sentence?

This seems to me to be a misplaced modifier, also called a dangling modifier. The modifying prepositional phrase, "on the first day of class," is syntactically placed so that it apparently refers to "grade-grubbing." The mere positioning of the phrase leads to the confusion.

The more commonly seen version of this mistake occurs when the prepositional phrase is set off as an introductory dependent clause. For example, "Buried under the snow, George found an old boot." George wasn't buried under the snow; the boot was.

(Dan's thoughtful analysis of anaphor resolution seems helpful, but that generally is a matter of pronoun confusion — figuring out which of more than one referent is linked to a particular pronoun — not applicable to your example.)

(And if John Lawler weighs in on this question, he'll set us all straight.)

I've had frustrations with this too. I remember trying in vain to explain to someone else the other meaning of a warning sign; I couldn't, mostly because I lacked the right vocabulary.

Here's what I found: Finding the right antecedent to the sentence-ending phrase "on the first day of class" appears to be called anaphor resolution. If the phrase occured in the first part of the sentence, it would be cataphoric. (To refer to either case, there is endophoric.) The particular problem with

I mention that I won't tolerate [grade-grubbing] on the first day of class.

looks like Nearest Referent, where "on the first day of class" is linked to the closer "tolerate" than to the further-away "mention."