Just because an organization is tax-exempt doesn't mean that donations to that organization are tax-deductible.
The Internal Revenue Code lists dozens of types of organizations that are exempt from paying federal income tax. Some of these organizations can accept tax-deductible donations, but others can't. Our Lions Club recently tracked down specifically what kind of organization we were. Lions Clubs are 501(c)(4) organizations, which include other civic organizations. Donations to 501(c)(4) organizations are generally not tax-deductible. If we are asked (and sometimes even when we're not) we must tell donors that their donations are not tax-deductible.
Donations to organizations that are classified 501(c)(3) though are generally tax-deductible as long as the donor is allowed to deduct the donation. These include organizations that are charitable, religious, scientific, educational, literary, work to prevent cruelty to children or animals, or foster national or international amateur sports competition. Give.org has a good discussion of the tax implications of donations.
To summarize: Don't assume that your donations are tax-deductible. Ask the organization or track down the tax status of the organization to be sure.