Where in halachah does it say that one shouldn't answer amein to someone else's berachah, if they have food in their mouth?
This halacha is mentioned in Kol Bo (§24). The Kol Bo mentions two possible reasons1 for why someone should not answer amein with food in his mouth:
Just as a person should not recite a b'racha with food in his mouth,2 he should not answer amein to a b'racha if food is in his mouth.
It is dangerous to talk with food in one's mouth.
The Kol Bo prefers the second reason, and he reads the Y'rushalmi (B'rachos 6:6) as implying that one should not answer amein with food in his mouth due to danger.3
1 The Kol Bo mentions another interpretation of the Y'rushalmi (quoted in fn. 3 below) that a person who is eating will not be yotzei on another person's b'racha because he will be concentrating on eating rather than on the b'racha.
3 שאלו את בן זומא מפני מה בא להן יין בתוך המזון כל אחד ואחד מברך לעצמו אמר להן מפני שאין בית הבליעה פנוי א"ר מנא הדא אמרה אהן דעטיש גו מיכלא אסור למימר ייס בגין סכנתא דנפשא