Being a math major, and the frequency that this topic seems to pop up, I thought I would dig into it a little bit because there are still usually shreds of truth hidden amongst the false information.
TLDR: A heloc, under the right circumstances, CAN speed up the loan payoff for some people, however done correctly it will likely only shave off roughly 3-4 years off of a 30yr note. It is not some magic formula to cut decades off your mortgage unless you make significant overpayments, (in which case you wouldn't need the heloc in the first place).
The basic premise of "chunking" large amounts onto their heloc which people seem to promote is false. It does no good to trade 50k worth of mortgage debt at 4% to a 50k heloc at 6% that keeps revolving from month to month and takes a year to pay down that heloc before doing the process again. Simple math says that 6% > 4%. However, if instead of taking large chunks, if you take smaller chunks (equal to your paycheck) the process can work.
As an example, lets assume that a married couple has a total income of 10k/month, and keeps 2k in the bank for emergency funds. Lets also assume this couple gets paid once per month on the first of the month. Typically this would mean their bank account spikes up to 12k on payday, and then slowly starts to trickle back down to 2k as the month progresses and they pay their bills.
Instead, what they could do is set up their bills such that all of their bills except the mortgage, are due as late in the month as possible. We will assume the mortgage is due on the first, and all other bills, (credit card, utilities, cable tv ect) are all due on the 20th. Now they will open up a heloc with at least 10k available limit, and during that first month pay the full 12k that they have in the bank towards the mortgage. At this point they have an empty heloc, no money in the bank, but 12k paid towards the note. On day 20 when all of the bills are due, they will pay the all of their bills by tapping into their heloc. This will max out their heloc at 10k, which stays maxed for roughly 10 days until they get paid on the first of the next month. This paycheck goes to paying off the heloc in its ENTIRETY, any additional amount owed on your heloc that revolves into the next month is a bad thing. Now we are back to the original situation, where the couple has no money in the bank, and an empty heloc. Rinse and repeat this process for the next 26 years or so and the mortgage will be paid off, and then replace the original 2k that the couple had in the bank.
This process does not involve making large 'over payments'. The only extra amount that you pay would be the 75-100 dollars a year or whatever amount your bank charges for its annual fee for the heloc. This translates into about 7 bucks per month, which even if you didn't have a heloc, and instead paid that 7 dollars over your mortgage per month, it would only shave off somewhere in the neighborhood of 6 months of your typical priced mortgage. Where as the heloc method would likely save 3-4 years.
The reason this process CAN work under some circumstances, is two fold. First, the couple had 12k in the bank right after payday (2k emergency funds and 10k paycheck) which is wasted money that is neither earning interest, nor saving interest. Using the heloc method, this money is immediately put to use to pay down the loan, and using the heloc at the end of the month to pay bills. The other reason is that while the heloc rates are higher than the normal mortgage, the amount of time that the heloc is being utilized is less. Keep in mind that in this scenario, there is only 10 days out of the month that interest is being accrued by the heloc. So instead of the 10k being on your mortgage for the entire month at 4%, you are moving it to your heloc for 10 days at 6%, (which prorates into an effective 2% over the course of a full month). This means that the heloc is paying less overall interest even though it has a higher interest rate. These two reasons (while small reasons) slowly start to chip away at the mortgage and speed up the process ever so slightly. This process would save a few years off of a typical ~200k loan.
Obviously this entire method is dependent on the exact terms of your origianl loan, your heloc rates, how far back into the month you can push all of your bills etc. Is it worth going through the hassle of setting all of this up to wipe out 3-4 years of mortgage, or the possibility of a heloc being frozen? That is up to you.