We have received a Property Tax Bill from the County of San Bernardino, California, stating the as the Lessee of Copiers we are required to pay property taxes on them - that the Lessor is not responsible for the property taxes. This is the first time we have ever been billed such property taxes. Is there a specific Property Tax Exemption Form that we should be filing?

As a California Special District, are we subject to Property Tax on Leased Copiers?
Answers
You should talk with your local CPA or Law firm.
Otherwise talk to the San Bernardino Tax assessor's office.
I would also look at your lease agreement and the bills from your leasing company to see if you are being charged for the property taxe.
You've raised multiple questions that require fact gathering. Issue #1 is what the lease agreement says about property tax responsibility. Most leases specifically assign liability for sales, use and property taxes to the lessee. As Mr. Baikie suggests, read (or ask the District's attorney to do so) the copier lease agreement.
Does the monthly bill from the lessor break out separate amounts for equipment rental, sales/use tax and personal property tax? Distinguish between the party that incurs the property tax obligation and the party that actually remits the payments. Lessors sometimes want to control the actual payment of taxes in order to confirm the payments are timely made.
Also read the other terms of the lease. Is the lessor is intended to be the owner of the equipment? In such cases, the lessee likely has either a fair market value purchase option or no purchase option and must return the equipment at the end of the lease. If the District can purchase the leased equipment for some nominal amount or gets significant rental credits to offset a purchase, the District may be deemed to own the equipment for tax purposes (i.e., the financing was really a lease intended as a sale). Again, a knowledgeable attorney should be able to guide you.
It would be worth a call to the SB tax assessor's office to find out what change triggered the issuance of a bill. Generally, taxing bodies want revenue and will try to find it from any party they can.
I don't know whether the District is entitled to relief from property tax as a quasi-government entity. The District's attorney should have the answer. If the District, as user of the equipment, is not liable for personal property tax, the lessor can likely claim the same exemption.
Ask to your leasing company or consult to any law firm.