If you can't get anything off the landlord, a household bill in your name (ESB, gas, UPC etc) in your name would be enough to prove your residency in the new place. If none of those will be in your name, try bringing in a bank statement with your new address on it. The OP will not qualify for rent allowance. As Berni said, you have to have been renting privately for a least 6 months before you qualify for it. But as the OP never mentioned it in the first place I imagine its not a concern for them.
If the landlord is not paying tax on his rental income, that is an issue between him and the tax man. His tenants have nothing to do with it.