Can you get hiv from sitting in a chair with dried blood if you have an open cut?

Theoretically: HIV is transmitted via blood communicating with another's blood, but in the scenario you described,with dried blood on a chair, transmission of live virus would be very unlikely.

No: Please forgive me. I find this scenario far-fetched, but I do believe that something happened and you must be worried if you've asked. Casual contact as might occur between family members not involved sexually or sharing needles will not transmit HIV. You have a right to be tested for HIV at minimal cost by your physician any time you so request. Best wishes.

Never been reported: HIV does not survive outside of the body for very long. If the blood was completely dry (and indeed contained HIV) the virus would almost certainly be inactive. To my knowledge, transmission via the example you describe has never been reported.