Well, with IRA funds, you wouldn't be paying the mortgage with those payments, as you are penalized substantially if you do anything with those funds unless specified exceptions prior to retirement.
Your neighbor, in theory, is correct - however, it is a lot more complex than he would make it seem.
With an equity line, it is likely that the rate is going to be variable, and if that is the case, it pretty much shoots down his plan right there. If it is a fixed rate, it makes it a bit more feasible, but you should consider talking with a tax advisor to see if it is beneficial to you.
Paying down the mortgage will result in a lower payment and more equity, basically. If you take out additional funds to invest, you want to make sure that the amount of the mortgage payment + tax benefits received is greater than the amount you would earn in interest - taxes applied for income and/or capital gains.
It would be extremely difficult to do in the short term, as I don't think you'll be able to find a short term (3 month) CD that has a rate anywhere near what you would need to get to even break even. In fact, I would actually be surprised if you find anything less than a 10 Year that would even allow you to break even - but keep in mind I live in Wisconsin, so the market may be different around here...
If you have a free and clear house, you may want to look to other options to gain income from the property. Fixing it up and selling it, for one. Or, you and your wife could possibly live in it. Another option you may want to take a look at is renting it out. In most cases, having a rental property will beat the pants off of any Certificate of Deposit you'll find in return.
If you are considering selling it right away though, you may want to consider if fixing it up is going to get you additional profit. Around here, at least, if you aren't able to do a substantial amount of work yourself, it is likely that you will not be getting an increase in value dollar for dollar the amount you put in.
Hope this helps - If you have any more questions, just let us know