Green’s Dictionary of Slang

furburger n.

[fur n. (1) + play on SE hamburger]

1. the vagina, esp. during the act of cunnilingus when it is ‘eaten’.

[Aus]B. Humphries Barry McKenzie [comic strip] in Complete Barry McKenzie (1988) 94: If youse get jak of stropping the Mulligan and feel like [...] tucking into a nice fresh fur-burger, tell the tart you love her.
[US]B. Rodgers Queens’ Vernacular 139: The vagina when erotically licked. [...] furburger.
[Aus]B. Humphries Traveller’s Tool 131: Once a little sheilah from the typing pool of the Russian Consulate in West Berlin offered me a nibble of her furburger.
[US](con. 1960s) G. Washington Blood Brothers 56: Boy was she grateful that night, and we had our dinner in bed (fur-burgers).
[Aus]G. Seal Lingo 126: Cunnilingus of various kinds may be referred to as a sixty-niner; a furburger; to go down, and muff-diving.
[NZ]McGill Reed Dict. of N.Z. Sl. 83: furburger and Y-bone Sexual meal, playing on the notion of a T-bone steak.

2. a hairpiece, a wig.

[SA]P. Slabolepszy ‘Over the Hill’ in Mooi Street (1994) 66: Ou Moerdyk’s getting himself a hairpiece next week. As long as he doesn’t play in the damn thing. Oke gives him a high-tackle – he’s left with a fur-burger in his hand.

In phrases

eat a furburger (v.)

(US) to perform cunnilingus.

[US]E. Folb Runnin’ Down Some Lines 236: eat a fur burger Engage in cunnilingus.
C. Woodard-Lynk Between Secrets 8: ‘I think she likes you girl!’ I said jokingly. ‘HA! Whatever! I’m strictly dickly and I don't eat fur burgers!’ .