If a co-tenant on a property (answers would preferably address residential cases, but commercial cases would be acceptable also) pays more rent than another, are they legally entitled to, for instance, more income from the sale of shared belongings? Would they be legally liable for more, if the tenants were penalised for some private nuisance?
Tenants, where there are more than one, are jointly and severally liable for the payment of the rent and are similarly jointly and severally liable for any laws that "the property" breaks.
Joint and several liability means that they are each individually responsible, and the landlord or anyone suing for damages may pursue any or all for satisfaction of the debt (they may go after one if, for example, the others cannot be found). How the tenants settle this between themselves is independent of the lease.
If the tenants have an agreement to split the rent in any proportion (including equal shares) then this may amount to a contract. However, family, domestic, social and voluntary agreements are presumed not to be intended to legally bind the participants. Whether this presumption would be overturned would depend on the specific facts - the existence of a written contract being a good one.
Notwithstanding, it is difficult to see how paying more rent on a property would entitle you to a greater share of another, different piece of joint property.