If we bought a business and the seller had a debt with the landlord, are we responsible for any debt contracted by the seller with the landlord?

Depends upon what your purchase contract says. If the contract is silent and/or you did not know of the debt then the contract can be voided as it was indirectly including terms you were not aware of.

It depends on the sales agreement and whether your purchase of the business all liabilities as well as the assets of the business.

If you bought the stock of the company that incurred the debt then you may be liable. If you only bought the assets then probably not. The buy sell agreement should address this situation. You should consult legal counsel.

Maybe. Did you buy stock or assets. If stock, then definitely. If just assets, you are not legally liable. If you bought the LLC, that would be the same as buying stock. However, if you bought assets, did you buy the remaining portion of the lease? If so, you probably bought his interest in the lease, meaning the past dues. Finally, assume that you only bought the remaining portion of the lease. If the landlord did not consent, then he is not obligated to honor your agreement and can demand that you pay the back amount to stay there. If you did not talk with an attorney before you bought the business, talk with one now. If you did consult with an attorney, ask him or her about the issue.