As a NY state landlord with issues that my tenants are abusing the property, but have no hard evidence, I want to evict my tenants. I see the damage, but can't blame them except by circumstantial evidence. Am I required to have tangible cause to give 30-day notice or go through the formal eviction process?

As many places in NewYork have rental control boards then you may be subject to specific rules.

There are some different rules in NYC than in the rest of the state. However, the main question here is whether there is a lease. If the tenants have a long term lease, you will need to show cause to end it early. If they are month to month, then no, you can terminate the tenancy with proper notice. However, if they refuse to vacate, you will have to go through the formal eviction process.

What type of abuse, the house is under they care and control,

As for as cause it really depends on exactly where you are at, and local rules.
I will sound a little mean here, that is something you needed to know the day before the first renter ever moved in. Plus have all attorney approved forms ready prior to ever renting, for eviction.

So what type of damage, is it on going,