This common malaphor (check the hits on google!) is used when someone is wanting to say “I had to bite my tongue” (struggling to not say something you really want to say). My guess is that the speaker is also thinking of “I showed my teeth” (displaying anger) or possibly “sink your teeth into” (become deeply involved). The latter is probably more likely as the words bite and sink are four letter words and are active verbs. Also, the tongue and teeth are near each other and so this adds to the confusion. Finally, teeth bite and tongues don’t so the mind might be trying to correct itself?