It seems that if such a word exists it would be rather useful. Often, whether listening to to certain people or talking with friends, I feel that certain ideas are willingly exaggerated for a certain effect--perhaps to elicit a "wow" response. But instead I am left feeling as though the deliberate exaggeration of the point was exactly what turned me off.
Hyperbolize means to use hyperbole; to exaggerate. As you may already know, the noun form, hyperbole, means obvious and intentional exaggeration. So you could say to your friend, "You are hyperbolizing, and I'm done listening now."
If they are repeating themselves to make a point, you could use the phrase ad nauseam. (I can't think of a verb form for that.) According to Wikipedia, ad nauseam describes an argument that has been continuing "to [the point of] nausea." If you say, "This has been discussed ad nauseam," it means the topic in question has been discussed extensively, and that everyone has grown tired of it.
And finally, in debate terminology, there is a logical fallacy known as argumentum ad nauseam (argument to the point of disgust, through repetition). (I can't think of a verb form for this either.) The American Heritage Dictionary defines this as follows:
Argumentum ad nauseam, or argument from repetition, or argumentum ad infinitum is an argument made repeatedly (possibly by different people) until nobody cares to discuss it any more.
To over-dramatize is to exaggerate for dramatic effect. And over-dramatization can certainly kill enthusiasm.
The other word that comes to mind is belabor which is excessive insistence of an argument even after the point is made.
Coupling these two may be overly dramatic but:
People tend to get annoyed when I belabor my over-dramatized arguments- I don't know why...
I've heard oversell.
"It's a dessert topping and a floor wax -- and it cures cancer!"
"I think you're overselling a bit."
Overpraise would work, too.
To labour the point (or labor) means:
to try too hard to express an idea, feeling or opinion, repeating it unnecessarily
according to the Cambridge Dictionary.
I think some synonyms for exaggerate can carry negative connotations, depending on the rest of the sentence, including: overstate, inflate, and embellish.
The best I could come up with, though, is the term self-aggrandize.
Overplay sounds close, though having never actually seen it used, I wouldn't know the precise connotation.
People here keep suggesting “hyperbolize” or “hyperbole” however, while hyperbole does mean to exaggerate a point, it does not mean to exaggerate a point to the points detriment. The connotation for the term “hyperbole” is far more positive and usually refers to the intentional exaggeration of a point as a means of underlining, reinforcing or proving a given point. The term “hyperbole” would not serve well for your intended use.