Green’s Dictionary of Slang

make like (a)... v.

1. [late 19C+] (orig. US) to imitate, to pretend to be, to behave like or as if; thus make like a chicken, (cf. make a noise like... under noise n.1 ).

2. [1950s+] (US, also do a..., imitate a...) as part of a number of phrs. all of which mean ‘go away’, ‘get lost’, e.g. make like a fart and blow away, ...dragster and lay rubber, ...drum and beat it, ...banana and split, ...cow pat and hit the trail, ...paper doll and cut out, ...tree and leave, ...rubber and roll on [an extensive selection can be found in Maledicta IV, 1, pp.40–42].

In phrases

make like a baby and head out (v.) (also make like a foetus and head out)

[1980s+] (US campus) to leave.

make like an alligator and drag ass (v.) (also make like horseshit and hit the trail) [pun on drag ass v.]

[1950s+] (US) to leave.