What is the ruling of Islamic law regarding a tenant's claim to have carried out renovations to a shop for a certain amount without proof, especially after the renewal of the contract without mention of any dues owed to him, and is this claim to be considered after the landlord's death?

Religious courts are the arbiters in such matters. The tenant is not permitted to refuse to vacate the premises after the contract expires. If he claims a right over your father, he must prove it with evidence or by your acknowledgment. Your father's liability must be cleared, and the rightful claimant must be given his due before dividing the inheritance among the heirs. If someone has no evidence to support his claim and you do not know the truth of it, then the oath is directed to you, and you swear by Allah that you have no knowledge of any right he has over your father, and his right will then be nullified. The heir swears an oath regarding the deceased's debt, based solely on his knowledge that he is unaware of any right the claimant has over his predecessor.