Verb (5-dan, む, transitive)
1. to thrust (something) into (something); to cram; to stuff; to shove
彼は左手をポケットに突っ込んでいる。His left hand is shoved in his pocket.
Verb (5-dan, む, intransitive)
2. to plunge into; to charge into; to rush into; to ram into; to crash into
奴らは真っ直ぐ突っ込んでくるだけだ！They'll just charge straight at us!
3. to delve into (a matter); to go into depth; to get to the heart (of something)
あまり深く突っ込まないでおこう。Let's not delve too deeply into it.
Verb (5-dan, む, transitive)
4. to press (someone) about; to point out (e.g. an inconsistency); to question sharply; to grill
何故あそこにいたのか突っ込まれるとまずい。It would be bad if they grilled me about why I was there.
5. to get involved with; to meddle in; to poke one's nose into
Verb (5-dan, む, intransitive)
6. to riposte; to retort; to quip
俺はどうしても一言だけ突っ込みたかった。I just had to get in one quip.