"Bogart" means either to hog, to take more than the fair share of something (particularly a joint), or less often, to bully, act tough and to be belligerent. Both of these usages come from the name of the actor Humphrey Bogart (1900-1957) who generally played tough-guy movie roles. The second usage makes sense to anyone who's seen "Bogey" on film, but how did a guy who only smoked tobacco on screen become the verb for hogging marijuana?

A subscriber of A.Word.A.Day named Chris Strolin offered this explanation:

While "to bogart" now means "to hog or take more than one's share," (largely because of the song lyric "Don't bogart that joint, m' friend. Pass it over to me again.") this is only because so many people have gotten it wrong for so many years. One of the great misunderstood slang phrases of my youth, "to bogart a joint" originally meant to dangle a marijuana cigarette from your lips in the same manner Humphrey Bogart would do with a regular cigarette while eyeing a sultry Lauren Bacall from across a smoky bar room. With the lit end pointing downward this way, the joint would burn much more rapidly and, worse, would do so to no one's benefit. . . . the social crime involved was not in hogging the marijuana but in wasting it!

(The song he quotes is "Don't Bogart Me" {also known as "Don't Bogart That Joint" after the repeated chorus}, written by Elliot Ingber and performed on the soundtrack of the 1969 movie "Easy Rider" by Fraternity Of Man (as well as Fraternity of Man's self-titled first album in 1968). It was later covered by Little Feat among others. )

This explanation made sense to me, so I went looking for confirmation -- but I found several completely different explanations from others. Word Detective users offered these other possibilities:

'I thought "bogart"ing something was to toss it away before it was done. As in, don't toss away that joint before I have a chance to get high . . . I was told it was a reference to the way Humphrey Bogart, as the tough guy, would always dramatically flick a half-finished cigarette when he was making a point.'

'Originally, I thought "bogarting a joint" meant to accidently soak the end in your mouth as often happened when unfiltered cigarettes (the kind Bogart smoked) stayed in your mouth for extended periods.We referred to it, in the 1950's, as "mugging" the cigarette.'

However, one person came up with a movie reference:
'In "The Roaring Twenties" Cagney is sharing a cig with Bogart. He keeps it. Hence, bogart - to hog.'

According to the U.S. Census Bureau (via dictionary.com), "Bogart" is also a town in Georgia, USA:

Bogart, GA (town, FIPS 9068)
  Location: 33.94731 N, 83.53232 W
  Population (1990): 1018 (424 housing units)
  Area: 6.5 sq km (land), 0.0 sq km (water)
  Zip code(s): 30622
And one user of urbandictionary.com actually claims the hog/steal usage comes from this town rather than the actor!
"to steal or borrow in a white trashy manner. Coined after the citizens of Bogart, GA."

So, barring a lot more scholarship and sources that pre-date the 1968 song, the exact route this term took to its present meaning will remain unsure.

Sources:
http://wordsmith.org/words/bogart.html
www.wordsmith.org/awad/awadmail107.html
http://www.virus-bs.ch/lyr/bogart.htm
http://www3.clearlight.com/~acsa/introjs.htm?/~acsa/songfile/DONTBOGA.HTM.
http://www.lyricsplayground.com/alpha/songs/d/dontbogartmedontbogartthatjoint.shtml
http://www.wordwizard.com/clubhouse/founddiscuss1.asp?Num=3900
http://www.urbandictionary.com/define.php?term=bogart
http://www.wordorigins.org/wordorb.htm