On the album Pornographitti by Extreme, released in the late 80s/early 90s, there is a song called "When I'm President".  It is a rock song with rapped lyrics.  After the guitar solo, there is an interchange between the singer and the guitarist        

Singer:  "Go ask Alice."        
Guitarist:  "What did he say?"        
Singer:  "Remember, I want to be elected."


They Might Be Giants - "Why Must I be Sad" (from the album 'John Henry')

"Why Must I be Sad" is sung from the perspective of a kid who hears all of his unspoken sadness given voice in the music of Alice Cooper. Alice says everything the kid has been wishing he could say about his alienated, frustrated, teenage world. It's another one of our speculative, non-autobiographical songs which uses the word "I" a lot.
(They Might Be Giants fan newsletter circa November 1995)

"Why Must I Be Sad"

No More Mr. Nice Guy
I Love The Dead
I've Been thinking about it
Now I understand what he said
Ask Me now
I understand the words that Alice said
I kick the rocks beneath me
I squint at the sunset sad sad sad
Why must I be sad?
The rows of dandelions growing all around me
Why must I be sad?

No one knows these things but me and him
so I'm writing everything down in a spiral notebook
in the hopes that one day
other people will feel as low as this
Welcome to my Nightmare
Dead Babies
Raped and Freezin'
You Drive Me Nervous
Elected
Generation Landslide
Under My Wheels
Muscle of Love
School's Out
Only Women Bleed
Billion Dollar Babies.
(This last part where all the song titles are listed, one person is singing the titles while another is singing the line "Why must I Be Sad?" over and over again)


"Scared" by Dangerous Toys. It even says so in the credits "...dedicated to Alice Cooper" Apparently the lead singer is a big Alice fan. The lyrics are as follows:

Scared

Who is the man in the white vest?
I'm getting nervous, becoming a mess
Is he the doctor's man to take me away?
Or the same damn demon from yesterday?
Waking up is virtually impossible
Something in this story won't let me go
You know I'm shaking and I'm feeling sick
Guess I'm gonna need a vodka quick
Momma's lace and whiskey on the top shelf
Somehow they got caught up in my dream
Monsters on my right and cobwebs on my left
Scary man standin' right in front of me
Hey man, I think I like being scared
An' I wish you all were there Tomorrow, if I get my strenght
I'll put my daddy's gun to use
I hate it, I hate it feels like I'm drowning
The man's tied me up and I think he's frowning
Black widow is his favorite bite
Why does the creature cling so tight
Makes her crawl on my face and neck
God, the man drives me a nervous wreck
Why am I starting to like this?
You know he hasn't hurt me yet
But it's different now, just a whim
Maybe I'm becoming his pet
Hey man, I think I like being scared
An' I wish you all were there
Man I really, really think, really think that I think
I like being scared

Other Mentions

Another track to mention is The Scorpions` 'Speedy's Coming' with the line "Do you like Alice Cooper, do you like Ringo Star"

Also Dr Hooks song "Everybody's Making It Big But Me" features the lines "Well I hear that Alice Cooper's got a foxy chick To wipe off his snake, and keep him rich"

'Geared & Primed' by Royal Court of China has a Alice Cooper mention as well in the line "I had a vision, I was Alice Cooper"