I'm not an American citizen but I was here as a student. I'm Italian and my international driver's permit expires in April. I came here in April 2015 and for 20 days I'm uncovered. What happens to me if I'll be stopped and my IDP is expired?
The expiration date that counts is your normal driving license. The IDP is nothing but a standardized translation of your license. As long as your Italian license is still valid, you are OK.
See How can I determine if I need an International Driving Permit? - it's generally only required when the script is different to yours.
Source: NZ citizen, and have used a NZ license without an IDP in the USA to rent and drive cars on several occasions, and have even been checked by police.