Landlord Missing - can tenant acquire title?

you say you are paying rent for the last 60 years. To whom you are paying rent. If nobody is claiming rent just enjoy the property. When somebody comes then you ask for his rights, how he is claiming title, then only your claim can be decided. Otherwise you can go on enjoying it only. 

Tenant had been paying rent to the landlord until about 25 years back when the landlord went missing and since then no rent is being paid to anyone.

The question is not of enjoying property which is not one's own. We are exploring avenues how to get legal title over the property, before some frivolous person makes a claim.

you have no choice at present. unless someone comes with a claim then only you can defend it. Otherwise you can get the revenue records changed and then based on that execute a settlement deed in favour of the family members.

Mine is a similar case. You mention that tahsildar should change the title in favour of the tenant. How can that be done ? Are you saying that land revenue / Non agricultural land tax receipts -if they are in the name of the tenant they can be used as proof ? Should a suit be filed to claim adverse posession ?

Based on the land tax and other revenue receipts for so many years you ask for a declaration of title from the court. You can also claim the same on adverse possession.