If a written contract is signed and dated by both partners does the written contract has to be witnessed to make it binding?
No, but some contracts must be notarized, most common being contracts for sale of land, but there are others.
Only a few kinds of documents must be witnessed.......the most common one coming to mind is a will....but even there an exception exists for so-called "holographic" wills which are entirely in the handwriting of the testator, dated and signed. I cannot think of any kind of contract that requires witnesses, but I suppose a party to a contract could make witnesses a condition of his/her/its participation in the contract.