I don't give a dirty handkerchief

Means the same as, "I don't give a flying fuck", "I don't give a Funyun", "I don't give a rat's patootie", etc.
This is a clean way of saying it -- though that hankie really needs to go in the wash. :-)

WarGrowlmon: What? Beelzemon?

Taomon: What are you doing here?

Rapidmon: Looking to hit us while we’re down, huh? That’s your style right?

Beelzemon: Hey, trash me if you want but I swear I’m here to help.

Rapidmon: Do you even know the meaning of the word?

Beelzemon: I’m on the level fellas, really.

WarGrowlmon growls at him.

Beelzemon: But I don't give a dirty handkerchief if you believe me or not, cause my partners believe in me.

Flashback to Mako giving Impmon his gun.

Mako: You can blast all the bad guys with it.