I have money in a savings account, so I received a 1099-INT form with the interest in Box 1, but no Box 3 was on it. Does that mean I enter $0 in Box 3 on my tax form?

Yes, if there is no information being reported in box 3, you can simply enter zero in the space.

Per 1099-INT instructions, 

Box 3. Shows interest on U.S. Savings Bonds, Treasury bills, Treasury bonds, and Treasury notes. This may or may not all be taxable. See Pub. 550. This interest is exempt from state and local income taxes. This interest is not included in box 1. See the instructions above for a taxable covered security acquired at a premium.