“Burlesque”, a song by Family, is about a bar in Leicester, England. Don’t know if it’s there anymore.
King Missile has breakfast at Kiev in Detachable Penis.
Laraby’s Gang by the Spin Doctors mentions “the story that the Nightingale sang (now now)” which I assume is a reference to the Nightingale Bar.
Cole Porter, “Well, Did You Evah!”:
She got pinched in the Astor bar
refers to the bar in the long gone Astor Hotel (Times Square, NYC)
Well we take our breezes and our dunes as they come up here. I tried.
Night Owl Blues - James Taylor
Puttin on the Ritz - Irving Berlin
Lifter Puller used to reference a Chinese restaurant called The Nankin in several of their songs. It’s closed now but it did exist at one time in Minneapolis, where the songs were set.
I think he is probably going to be the king of this.
Napoleon Pizza House is a real place and is mentioned in one or two songs.
The Andrea True Connection, the disco group that had a hit with “More More More,” had a lesser hit called “N.Y., You Got Me Dancing,” which references The Barefoot Boy, 12 West, and Regine’s, which were all disco bars/clubs at the time.
Concerto For Horn And Hardart 
“At the Copa, Copacabana…”
*Well let’s all go, down to Dumas Walker
Let’s all go, down to Dumas Walker
We’ll get a slaw burger, fries and a bottle of Ski
Bring it on out to my baby and me *
“Dumas Walker” by The Kentucky Headhunters
This is sort of a combination of two real places.
Well, I pawned my Smith-Corona
And I went to meet my man
He hangs out down on Alvarado Street
By the Pioneer Chicken stand
“Carmelita” by Warren Zevon
Pioneer Chicken is a real chain but I don’t know if there was actually one on Alvarado St.
I am surprised this hasn’t been mentioned, because it’s the first thing I thought of when I read the OP.
In the song Carey by Joni Mitchell, Joni mentions a bar called the Mermaid Café. It was in Matala, on the island Crete:
Come on down to the Mermaid Café, and I will buy you a bottle of wine.
And we’ll laugh and toast to nothing and smash our empty glasses down.
Gotta love Joni…
Does a telephone number count?
Many big band names played in the Hotel Pennsylvania’s Cafe Rouge in New York City, including the Glenn Miller Orchestra.[1] The hotel’s telephone number, PEnnsylvania 6-5000, inspired the Glenn Miller 1940 Top 5 Billboard hit of the same name
You’re the top, You’re a Waldorf Salad
- Cole Porter, You’re The Top.
Searched the thread but I may have missed it:
Walking in Memphis, by Marc Cohn mentions the Hollywood (Cafe).
Now, Muriel plays piano
Every Friday at the Hollywood
And they brought me down to see her
And they asked me if I would
Do a little number
And I sang with all my might
She said, “Tell me are you a Christian, child?”
And I said, “Ma’am, I am tonight!”
Also, American Pie mentions The Levee
Up and down the City Road
In and out The Eagle
That’s the way the money goes
Pop goes the weasel.
The Eagle was a pub in City Road, London.
“Puttin’ on the Ritz”
More a nightclub than a bar, but Studio 54 gets mentioned in Chic’s “Le Freak”: “so come on down to 54/find a spot out on the floor. Awww… Freak out!”
Back during the second world war…“I Left My Heart at the Stage Door Canteen.”
From the early 60s
Sugar Shack…Now, I know this is a real place. I have been there.
The Name of the Place Is * I Iike It Like That*. I don’t know if this is a real place. I have been (by people I kind of trust) told it is, however.