If you're in an own door premises, you are normally liable for the bin collection charges, and standard tenancy agreements usually provide for this. Obviously, that does not include any liability in respect of previous tenants, but in fairness to the landlord he didn't incur the charges either. Local authorities have a bad habit of treating waste collection charges rather like the rates of old, and attaching them to the address rather than the individual in order to try and secure payment. To the best of my knowledge, there's no legislative basis for them to do this, and it seems inappropriate to remove your service on the basis of someone else's arrears.
What I would suggest is that you ask the landlord to write to DCC explaining when your tenancy commenced and confirming that you cannot therefore be liable for any previous charges outstanding. If they wish to secure payment they can then either attempt to pursue the landlord for them, or track the previous tenant.
I'd write to DCC as well either -
(a) informing them that you are happy to pay your charges since moving in and requesting restoration of service (and, if there was a reason for non-payment, such as disputed bill from the beginning, explaining that bit), or,
(b) if you have paid your charges since moving in, telling them that the withdrawal of service is unreasonable and demanding for its prompt restoration.