It's a little complicated, but I'll try to help you out. I'm not a CFP nor an CPA, but I do manage our investments.
First of all, the tax benefit you get from your mortgage is that you can take the mortgage interest paid + property taxes as deductions from you federal income taxes when you itemize on your taxes and do not take the standard deduction.
(BTW - knowing you're in Missouri doesn't help because each county/township has different rates of property taxes.)
So, let's assume you have a mortgage of $250k. That means that you will pay about $15,000 in interest on a 30 year fixed loan of 6% (about the going rate). The standard deduction is $7300, so you've already beat the standard deduction and you should itemize. If you pay an additional $5000 per year in property taxes, then you have a total of $20,000 you can claim as a deduction on your taxes to get you adjusted gross income (AGI) down as low as possible. Your marginal tax bracket is based on how much tax you pay on your last dollar, so this is very important as it can lower your marginal tax bracket. Lower tax bracket = less money paid in income taxes AND how much you pay in short term capital gains.
But let's say you are STILL in the highest marginal tax bracket. That is that you are married, filing jointly and earn > $336,550. You are in the 35% marginal tax bracket.
To make it very, very simplified, by itemizing, you are saving 35% of $12,700 or $4445. If that was the only thing to think about, then it wouldn't be worth it to carry a mortgage just for the deduction. It turns out the deduction is just a small perk because the REAL return comes in investing that money you would pay on your mortgage.
Since its inception, the stock market has earned about 11% annualized. That means that over the long haul - 15, 20, 30 years - you can be relatively sure to gain about 11% on your investments before taxes. After taxes and fees about 8.5% - 9% is a realistic number assuming you chuck all of your money into index funds (a passive investment strategy) that earn what the stock market indicies would earn. Already, if you hold a 6% mortgage, you're gaining 2.5% - 3% per year over that. Moreover, assuming you don't cash out the investments within 12 months of the investment, you will only pay 15% taxes on capital gains... NOT YOUR MARGINAL TAX RATE! You only pay your tax rate if you cash out and take short term capital gains.
Where it gets complicated is that if you invest in equity mutual funds, that fund manager is buying and selling stocks within that fund (this is called "turnover") and the higher the turnover rate, the more short-term capital gains taxes you will pay on your investments. I hope that makes sense. When a fund manager buys and sells within the fund, you will pay capital gains taxes on that. You will also pay capital gains on dividend distributions within the funds. That is why when you are choosing funds for your portfolio, it's important to buy "tax-efficient" funds if you are in this higher tax bracket.
So by carrying your mortgage you are getting the tax break based on your marginal tax bracket, plus you are gaining about 3% or more annually by investing the money in mutual funds (I don't do stocks at all, as I know I cannot beat the professional fund managers returns you get in mutual funds). But it boils down to what funds you actually hold and their tax efficiency that will give you your final numbers.
For actual figures, talk to a CPA this spring when s/he is doing your taxes and they can give you a concrete number based on your taxes and investment level. For us, I estimate a net gain of about $10,000/year for carrying a mortgage.
If you are going to hold your money in cash and you don't want to invest it, then the above is all moot and you should just pay off your mortgage.
