Of American-English origin, the informal phrase don’t hold your breath means don’t wait in anxious anticipation.

The image is of somebody holding their breath when anxious or excited about something—this is the image “it might hurt you” jocularly alludes to in the text containing the second-earliest occurrence of the phrase that I have found, from The Fort Scott Democrat (Fort Scott, Kansas) of Saturday 8^th June 1861:

Readers of the Democrat haven’t we presented you with a good number of this paper. We pause for a reply—but don’t hold your breath till you hear from us again; it might hurt you.
G. A. Reynolds.

The earliest occurrence of the phrase that I have found is from the beginning of Man Over the Falls, about a man whose idée fixe was to plunge over a waterfall, published in The American Citizen (Jackson, Michigan) of Wednesday 15^th November 1854:

Don’t hold your breath, dear reader, for he is entirely safe now, and survives to tell his own story—“But how did it happen?”—That is just the point where truth is stranger than fiction—for this event occurred not by any sudden accident, but of his own will, and that too notwithstanding the fact, that he had once previously taken the plunge over this awful cataract, and five or six years after had been drawn out the foaming waters far below, senseless, bruised, and bleeding.

The phrase then occurs in the following from The Appleton Post (Appleton, Wisconsin) of Thursday 13^th January 1870:

Don’t hold your breath waiting for Minnie Warren * to die, out there in California. The story of her expected demise arose from some joker’s assertion that she couldn’t live long.

* This probably refers to Minnie Warren (died 1878), an entertainer, affected by dwarfism, associated with the U.S. showman Phineas Taylor Barnum (1810-1891).

The phrase has often been used punningly—as in the following from the column Today in Queen’s Park, by Don O’Hearn, published in The Windsor Star (Windsor, Ontario, Canada) of Saturday 29^th October 1966:

Toronto — […] Improvement in air pollution control in this area is expected within three years. But don’t hold your breath meanwhile.

The phrase is also used punningly in this advertisement for the indoor air quality solutions provided by Munn’s, published in The Lake Sentinel (Orlando, Florida) of Thursday 9^th October 1997:

HOLD THAT BREATH

[…]
Our Healthy Indoor Living experts are specially trained to find a problem at its source and correct it. If your duct system isn’t contaminated, we’ll be the first to tell you. And wouldn’t you breathe a little easier just knowing? Give Munn’s a call today at 787-7741.
Don’t hold your breath any longer, just give us a call while you’re thinking about it at 787-7741. Thank you.