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 (either as a stipend or as a reimbursement of other costs) is taxable income. We will file a 1099-MISC form for anyone we pay more than $600 in a tax year: that could include you, your collaborators, or contractors that are paid 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. But if you buy something for your project and request reimbursement from your grant funds, that reimbursement is taxable income.
If you expect to be receiving a lot of your grant funds directly, you may want to include the added tax burden in your budget.