UK charity information:
No deduction for donation of items (except as noted by Guya), only cash.
If a museum, church, or other charity participates in "Gift Aid", then any donations you make can be listed individually on your return. You need to find out at the time of the donation. Museum annual memberships and entrance fees count, provided the museum participates in Gift Aid.
There are workarounds regarding donations of very valuable items. This is usually done via auction or sale; where the proceeds are then delivered to the charity. However, the proceeds are subject to UK capital gains tax, so the deduction is somewhat lessened.
There are also workarounds related to deductions for donations of items on a UK inheritance return. That is, if you die and leave your Van Gogh to a museum, the government will adjust your IHT tax accordingly. I don't do UK IHT, so I don't know how this works; as a member of the Art Fund, I know it's done through some sort of negotiation with IR.
Don't quote me as an expert regarding the workarounds! I most definitely don't know anything about it except that some mechanism of some sort exists in some fashion.
US charity information:
In the US, you do NOT get to dictate the market value yourself. You must use a realistic FMV. If the item is of high value, you need an appraisal. New laws passed this summer have made donations of items of minor value no longer deductible. These new laws also require receipts for all donations, no matter how small. Your church, if it doesn't provide a receipt for the collection plate, needs to start doing so now.
The IRS keeps statistics on everything. In 2003, the "Fair Market Value" of all those used clothes and goods came to $9 billion! Since this summer, you can't deduct them unless they're in "good" condition. The bill is specifically referring to items that have a low value, such as socks and underwear.