I sold a rental property. When does the new owner take responsibilities for the lease with the tenant?

One second after the closing. The minute the deed is transferred and the closing is done, it is now, the new owners house and you have nothing to do with it.

The new owner takes over the written lease and is then the landlord. The deposit paid by the renters, should have been shown in the closing paper work.

While I don't disagree with Chuck, this is a contract issue. The contract of sale can specify the effective date of the transition of responsibility. If it doesn't then immediately after the closing is complete.

To be totally clear about deposits, you must turn them over to the new owner.
If there are any other financial disputes ongoing, such as a promise by you to reimburse a tenant for something they fixed at their own expense, or an insurance settlement for a burst pipe, you can try to weasel out of them legally, but it would be unethical.
Not suggesting that is it. Just not sure what it could be.

Immediately once the whole process is being done and the home is now on the name of new person then its all belong to the new landlord.

Where is this property located?