You will pay less interest the faster you pay off principal. Once the principal is paid off, of course there is no more interest.
With every payment you make, the amount of interest decreases a little bit and the amount going to principal goes up by the same amount. The monthly payment does not change on a fixed-rate mortgage. If it does, it will be because property taxes and/or insurance are escrowed, and as costs for these rise, the payment will rise along with those costs.
In the early years, the payment is interest-loaded. In the final years, payments are mostly principal, you have already paid most of the interest you will pay. On a 30 year mortgage, the break-even point is somewhere between years 20 and 22. So, yes, the more you pay toward principal, especially in the early years, the faster the loan will be paid off.
However, it only makes financial sense to pay off a mortgage if:
A. You are nearing retirement and want the security of knowing the house is paid for.
B. The interest rate on the mortgage is close to or higher than the rate of return you can get on your money if you invested it instead.
If you know you can average 5% a year return on your money and the interest rate is 5% or higher, then every dollar you pay toward principal is a better investment. You are investing at that interest rate plus whatever tax savings you may get from deducting mortgage interest on your tax return.
But if you know you can get 5% or more investing your money over the long term, and your interest rate is below 5%, you're better off investing that money instead.
Studies have shown that the biggest portion of net worth that individuals and married couples have as they head into retirement is the equity in their home.. Much greater than 50%. Too many people are house rich and cash/investment poor.