Even if they know you are coming, there is no way they can clean up a place that they have caused damage to.

@Juan BustosThis is one of the great things about tax sales; they clean up title issues where people didn't do what they were supposed to. 

i just got a request...well more like a mandate, from the city....code inspector walked the apartment and found a few minor things..no biggie...the thing that pissed me off was he said the bathroom sink was filthy and i had to clean it...i said, if i have to start cleaning my tenants' sinks, they'll be getting 30 day notices...since when is it a landlord's responsibility to clean up after a tenant?!

Not the cheapest plumbing outfit out there, (I paid ~$200 for them to scope the pipes and do a general inspection and ~$160 to clean up a toilet and shower issue my tenant had) but you probably don't want to work with the bottom feeders anyways.

I know of a couple banks that do what is called asset depletion.

You should also require this information is updated annually.You can still place reasonable restrictions on the animal, such as: Requiring a current (and annually updated) Veterinarian report on the health of the animal, including vaccine status; current appropriate local licensing of the animal if required in your jurisdiction; require the animal be spayed or neutered unless contraindicated by the Veterinarian; require the animal to be in a carrier or on a leash when in common areas of Multi-Family properties, particularly elevators or hallways and stairwells; require tenants to clean up after the animal in common areas and/or within their unit, and prevent offensive odors from emanating from their unit; aggressive or nuisance animals (includes frequent barking that is NOT an alert by the animal) can be cause for removal of the animal and/or eviction of the tenants.

Take garbage bags and lots of old towels because oil tanks are never " empty"  In this situation, I'd probably opt to hold-back funds in escrow to pay for a "professional" to remove the tank, perform any cleanup and "certify" all is good (making the lender happy).  

I agree with Jimmy Klein. as a landlord I haven't had this issue. but as a commercial hvac contractor I have had issues with pipes bursting etc. you don't offer the tenant anything. immediately file a claim with your insurance company. hire a remediation crew to clean. up any if the water/ damage that occurred.

Based on the extra explanation, an asset depletion loan may work. 

Possibly you could use that as leverage to make the seller remove the tank and handle the clean up.