You are wise to be concerned with charitable contributions - something Americans are quite good at - we are generous. The purpose of income tax deductions for charitable contributions is to make it so the donations are made with untaxed, pre-tax dollars. If no dollars are taxed, all dollars are pretax. Hence there doesn't seem to be any reason that this would affect charities in the way you mention.
Shoot, only about 25% of filers itemize anyway - and they're the only ones who can take deductions for charitable contributions.
Under the FairTax, you pay no payroll taxes on charitable contributions. Therefore the FairTax would actually benefit charities.