If your grant is paid to a 501(c)(3) organization, or a project or organization which has a 501(c)(3) organization as your fiscal sponsor, the answer is no.
If your grant is paid to any other incorporated entity (like an LLC, if you have a business), it's the responsibility of that entity to calculate and pay any applicable taxes.
If those two situations don't apply, you're either an individual or you're with an unincorporated group. In that case, any grant funds that are paid directly to you as a stipend or payment for your work for the grant ARE taxable income. We will file a 1099 form for anyone we pay more than $600 in compensation in a tax year: that could include you, your collaborators, or contractors that are paid for their labor with your grant funds.
If the Foundation buys goods on your behalf for a grant -- we directly pay for equipment, hotel, airfare, etc. -- that does not have any tax implications for you. If you buy something for your work and request reimbursement from your grant funds, that reimbursement is NOT taxable income.
If you expect to be receiving a lot of your grant funds as a stipend paid to you, you may want to include the added tax burden in your budget.