Do I need to claim the full value of a deduction or tax credit (such as a charitable donation or foreign tax credit), or can I claim less?

However, if you need to provide a copy of your return to (say) a lender, they might question it.

Sch A deductions are OPTIONAL not required  and you are not required to claim the entire amount if you don't want to do so. 

OP also asked about foreign tax credit.

however, there was an interesting court case many years ago.  to avoid alternative minimum tax a couple under reported their deductible real estate taxes.  the court said ok for regular tax purposes but for AMT the full amount must be used. 

could never find out if there was an appeal and the decision reversed.