Technically, Taxes are not part of your "mortgage". But most lenders will retain an extra amount (that is, on top of mortgage payment) to pay for taxes, insurance, etc. To you, it will look like a single mortgage payment, but in reality you are paying more than that.
Homestead exemption will reduce your real estate taxes, so the net effect is that it should help reduce the payment you pay to the lender. But this is true only if this is the only change. If your insurance goes up, for example, then this increase may offset the reduction the exemption created. Still, homestead exemption is something you want, why pay more taxes?