The American Heritage Dictionary defines "dog it" as:
Do less than is required; loaf or shirk. For example, I'm afraid our donors are dogging it this year. This expression originated in sports and soon was transferred to other endeavors. [Slang; c. 1900]
Move slowly, as in We just dogged it along from California to Oregon.
Does anyone have more insight into the etymology suggested by the AHD? What are some early examples of the use of this expression?
As an aside, I originally thought there was a connection between the term "dog it" and the term "dog day" (since both seem to associate dogs with lethargy). However, this seems to not be the case. Wiktionary suggests an ancient origin for this latter idiom:
Calque of Latin diēs caniculārēs (“puppy days”), a calque of Ancient Greek κυνάδες ἡμέραι (kunádes hēmérai, “dog days”), from κυνάς (kunás, “of or related to dogs”), from Κῠ́ων (Kúōn, “the Dog”) in reference to the star Sirius, which appears in Homeric Greek as "Orion's dog". The return of Sirius to the night sky (its heliacal rising), occurring in antiquity around July 25 (Athens) or 29 (Rome), was considered by the Greeks and Romans to herald what were considered the hottest, least healthy, and least lucky days of summer
(See also the SE question on "dog days".)