Green’s Dictionary of Slang

beat one’s gums v.

also beat one’s chops, ...lip, beat on one’s chops, beat up one’s chops, …gum(s), beat up the chops, ...gums, bat one’s gums, chop one’s gums,flap one’s gums, …teeth, smack one’s gums
[SE beat + chops n.1 (1)//SE gums/SE lips]
[1930s+] (US)

1. to chatter, to talk, esp. in an irritating manner.

2. to talk in a melodramatic manner.

3. to eat.

4. to complain.

In exclamations

beat your gums!

[1950s+] don’t talk rubbish!