If the original contract does not have this functionality built in, the only way to do this is to transfer ownership of the original contract to a new contract that looks something like this:
contract LimitedOwner is Ownable
OriginalContract public ownedTokenContract;
// Explicitly allow these functions to be called
function burn(uint256 _amount) onlyOwner
function mint(address _address, uint256 _amount) onlyOwner
// All the original contract's owner-only functions that have not been
// implemented in this LimitedOwner contract can never be called again.
LimitedOwner should not be able to call
transferOwnership on the original contract, because then the disabled functions could be re-enabled by transferring ownership away from the
LimitedOwner. Instead, the
LimitedOwner contract should have its own
transferOwnership function to keep the ability to transfer ownership and to further limit the owner's abilities in the future.
If your original contract had no
transferOwnership (or similar) function, it would not have been possible.