If I have a small business (sole proprietorship) where I pay a monthly lease payment, and monthly taxes owed on property I lease. Do I have to file/provide a 1099?

Assuming you are leasing a commercial space, yes, you will need to issue a 1099-Misc to your landlord that reflects all payments made to the landlord.  However if your landlord is taxed as a corporation, then you do not need to issue a 1099-Misc.  

You will probably also need to obtain a completed W-9 from your landlord.   Regarding W-9s, the IRS has stated the following:

Use Form W-9 to provide your correct Taxpayer Identification Number (TIN) to the person who is required to file an information return with the IRS to report, for example:

• Income paid to you.
• Real estate transactions.
• Mortgage interest you paid.
• Acquisition or abandonment of secured property.
• Cancellation of debt.
• Contributions you made to an IRA.

About Form W-9, Request for Taxpayer Identification Number and Certification

@GCKirt