Logo descriptions by bigrene2 and Shadeed A. Kelly
Logo captures by Eric S. and Others
Editions by Curiousgeorge60
Background: This was a television production company formed by David Geffen as a division of Geffen Pictures in association with Warner Bros. Television. Nowdays, Beetlejuice is owned by Warner Bros. Entertainment, while Geffen Records (the parent company) is currently a subsidiary of the Universal Music Group. But Geffen sold Geffen Pictures to Universal Studios.
1st Logo
(1989-1990)
Logo: Superimposed in the credits, we have:

Produced in Association with
The Geffen Film Company
and
Tim Burton, Inc.
with a small Geffen logo on the side of the Geffen company name.
FX/SFX: None, it's superimposed in the credits.
Music/Sounds: The end title theme from Beetlejuice.
Availability: Can be seen on 1st season episodes of Beetlejuice on AOL's In2TV.
Scare Factor: It's based on how you feel about the music. It can range from none to nightmare.
2nd Logo
(1990-1991)Nickname: "Circular G"
Logo: On a blue/gray gradient background, we see a big version of the Geffen logo as in the movie logo and below it is the following words:
THE GEFFEN FILM COMPANY
IN ASSOCIATION WITH
WARNER BROS. TELEVISION
Variant: Later in the season, the WB shield appears on the right to join the Geffen logo and the logo now reads "A GEFFEN PICTURES RELEASE", "DISTRIBUTED BY WARNER BROS." and "A TIME WARNER COMPANY", stacked onto each other. Underneath is a copyright information. All placed on a purple (or blue/dark blue gradient) background.
FX/SFX: None, it's a still logo.
Music/Sounds: Only the Beetlejuice cartoon's closing theme finished.
Availability: Can be currently seen on Beetlejuice episodes on AOL's In2TV. Was last seen when Cartoon Network and Nickelodeon re-ran Beetlejuice in the mid-late 1990's. The Geffen/WB logo combo is extremely rare.
Scare Factor: None.