I assume the interest amount I'm currently paying for my 30-year mortgage is based on a 30-year amortization trajectory. If I refinance to 25 years, or even 15, what happens to the interest I've already paid? Is it 100% put toward the interest due in my new mortgage trajectory? Is any of it "lost"?
Nothing happenens to it.
You have a table of outstanding debt. THAT is refinanced. The past is the past. Your outstanding debt - not your original debt - plus whatever charge is there for refinancing minus any additional money you may put in is taken out as new mortgage.
What happens to the interest I've already paid? Is it 100% put toward the interest due in my new mortgage trajectory? Is any of it "lost"?
The interest you already paid is simply your cost to borrow the principal, none of it is put towards your new mortgage when refinancing.
In some ways, all interest is "lost" aside from some potential tax benefit, it's overhead. On the other hand, it enabled you to buy a house sooner than you'd otherwise have been able to.
Refinancing can affect the total amount of interest you'll have paid by changing the total time you're in repayment and by changing the interest rate you're paying. Refinancing to a shorter term typically only makes sense if you are also getting a lower interest rate (and it has to be low enough to more than offset the cost of refinancing). If not getting a lower rate, it makes more sense to simply apply extra payments than to refinance as @D Stanley mentioned.
Just as a savings account pays you interest each month based on the amount in your account, the interest you pay on your mortgage is the amount that you still owe (the principal) times the interest rate. Each month you pay for the use of the lender's money. When you pay off the mortgage (by refinancing or getting a windfall) that's the end of your payments, and the end of your interest obligation. If you refinance, that's a new transaction, with its own new interest rate, and the cycle starts again: each month you pay interest equal to the amount that you still owe times the interest rate.
The amortization tables that you look at simply show you how much you're paying in interest and how much you're paying against the principal. Each month you make payments that reduce the principal, and that, in turn, reduces the amount of interest that you owe the next month. Since the interest has now gone down, more of your payment goes to principal. That's why you see such a small reduction in the principal amount early on: most of the monthly payment goes to interest. And that's simply because you owe more money; as the amount you owe goes down, the monthly interest goes down.
Just a minor clarification: the monthly interest is based on the monthly interest rate. Mortgage interest rates are usually quoted on an annual basis, so you have to divide by 12 to get the monthly rate. A 6% mortgage would require .5% of the principal amount in interest each month.