I'm not sure when they open, but California Clipper might be the closest drinking establishment to Treat (haven't been there in a while, but never had a bad meal there, either). Great atmosphere, good selection, but a warning - some have critiqued their mixed drinks as not so well executed. They may still have music sometimes, I'm not really sure. (To be accurate: There is an old-school storefront bar on North a few blocks E of Kedzie, but there's nothing inviting about it, and I've never had occasion to enter.)
The Continental is a pretty generic 4 AM bar that is beloved of the fixie-and-facial-hair set, right next to the always excellent Feed. DJs and a late-night pick-up vibe. Chicago and California, I think. So probably second closest to Treat.
Tumbao is a fun Puerto Rican club/bar a few doors down from Streetside at Armitage/Kedzie. I think they have live music and/or DJs 3 or 4 nights a week. Great wing and beer special on Mondays, and run by cool people.
A few other places I like that are technically in Logan Square:
The Burlington is a faux dive, Whirlaway is a "real" dive - both attract the same youthful demographic, but there are generally pretty good DJs at the former and an excellent jukebox and old-school host at the latter - I forget her name, but the owner is very sweet and a real character - she's been running her joint since "81 or so, and is happy to tell a few stories of the bad old days. Good beer selection at both, but neither is terribly cheap.
Coles' (named for the owner, the bespectacled young guy usually behind the bar) is a new space just south of Whistler that does the usual, but does it well - good taps, pool table, fair prices, and often music or literary events in the back room.
The Two Way Lounge at Milwaukee and Fullerton is for some reason one of my favorite "dives" in Chicago - I dig the weird triangular layout, the very cheap drinks, and the often surly / harried bartenders. Kind of the anti-Whistler, in a good way.
Boni's Sports Bar has a 4 am license and late at night, will make any of these other places look like a kindergarten. Have a Long Island Iced Tea and try to remember the last time you were in an establishment that sawed off their rest room doors....