Or can I only get back what the charity can sell it for?
What if they use it rather than sell it ie donating old furniture to a shelter or an old TV assuming I still have the receipt or even bigger items like if i donate art work to a museum?
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In the UK the primary way for most people to claim back tax on charitable donations is through Gift Aid; this is only meaningful to you as an individual if you are a higher rate tax payer, as the charity claims back the basic rate portion of tax and you claim back the higher rate portion.
Gift Aid is only applicable to cash donations. If you donate physical goods to a charity shop, you can ask the shop to treat the money raised by the sale of those items as a cash donation and they will send you a letter stating how much was raised, which you can include in the Gift Aid portion of your tax return.
Additionally, if you donate land, property or shares to a charity you can deduct the fair market value at the time of donation from your taxable income in your self assessment tax return. The rules about this are here and an explanation of how to calculate the value of the donation is here. Note that all the examples of "property" imply that the term is used in the sense of houses / buildings / real estate rather than than simply "something you own". The helpsheets make this more explicit by referring to "land or buildings" and "real property".
In summary: if you donate old furniture to a shelter for them to use, you cannot claim back any tax.
In the USA, when you are claiming a donation of non-cash, there are 2 questions on the tax form that are important.
Usually you make an estimation as what the items are worth today. Most of the stuff you donate doesn't have a large value.
But since you have a receipt of what you paid originally, that takes some of the guesswork out of the equation.
While the UK tax laws are different, they should be somewhat similar in this regards.
Of course, you should consult with your tax consultant to make sure.
If the value is not very high, you are likely to not have much (or any) trouble with the tax people.