It is supposed to be originated in America and the oldest written records of the phrase can be traced from 1870s. For example in a book named 'Western Wilds, and the Men Who Redeem Them' written by John Hanson Beadle which is originally published in the 1877.
[ Examples ]
1. Do whatever you can, but don't bite off more than you chew.
2. By accepting two part-time jobs, he is clearly biting off more than he can chew.
3. It feels like I bit off more than I could chew when I promised to complete this worksheet in one day.
