When the company went out of business, the warranty was likely picked up by another company. Check your documents for an 800 number or other contact information for forwarding details.
A lifetime warranty is only as good as the company giving it. Since the company is no longer in business one can assume there are no assets to go after. So in realtiy unless someone from the dissolved firm takes pity on you and agree to perform your job you are out of luck and the money for the warranty.
It depends upon what happened to the business. If it simply closed, you are out of luck. If another business bought or absorbed it, you may have a claim against the new company.
If the furniture was made by a small time shop and it closed you are probably out of luck. However, if the furniture is a widely known brand you can just call their headquarters and the warranty should be good.
Obviously there is no remedy for this kind of situation and on anyway we cannot go ahead for further proceeding on this kind of business strategies. I think this may be the correct reasonable answer for your searching