We all want to save money, especially in these turbulent times. Your house payment has 3 components (if it's like most of world's): Principal repayment plus Interest, Tax escrow and Insurance escrow. We generally refer to the payment as PITI for each of these components.
Interest payment depends on the interest rate of your mortgage loan. Right now banks are paying only about 1% on bank savings accounts and charging about 4% or more for mortgage interest. This means from an investing point of view that paying down your mortgage balance is saving you about 3% or more in interest you don't pay.
Huh? If you kept a large balance in your savings account, you would be receiving only the 1% rate while you would be paying the 4% for the loan. So, by moving the money from savings to the loan, you would save the difference.
Long term interest rates both on savings and on mortgage loans may rise, but your mortgage interest is probably fixed at whatever rate (e.g. 4%). So, in later years the 2 accounts for savings and the loan may come closer to being in balance, earning as much as you're paying.
If rates on savings accounts rise above your loan rate, then you should have the money in your savings account and not pay down the loan. It is complicated. An accountant or CPA can explain these principles to you to help you decide what to do, especially since mortgage interest is deductible for Federal income tax purposes, and interest income is taxable.
As to insurance, you have to shop around to find homeowner's insurance at a lower rate to save money on insurance escrow. The escrow account merely collects the money from you and pays the bills as they come in. So, if you pick a lower insurance premium from a different company or different policy, you could save on that portion of your monthly payment.
Lastly, the real estate taxes are set by the taxing districts where your house is located and the escrow pays whatever is due each January. To lower the amount needed each month, you have to get the tax appraisal district to declare the value of your house is lower than it currently is.
The rates are set by the budgets of all those taxing districts. So, the only way to lower the taxes is to fight the budget increases of the taxing districts and/or the asessed value of the property. Every May you are allowed to protest the value and hopefully you can get your assessment lowered. Fighting budget increases is a difficult task, since you have to go to each hearing the taxing body holds and fight against budget increases.
Paying off your mortgage loan will not reduce your assessed value. But, as noted above, it could result in savings from reduced interest payments. I hope this helps clarify things for you.