Rather than focusing on minimizing expenses (for which lots of folks have already provided some great ideas), I like to think about the income side of the equation. It's never a good idea to be running purely on savings for long, if you don't have to.
One thing that I did while starting 15Five, in addition to minimizing all unnecessary expenses like office space, etc, was to do a small amount of high-value consulting. For example, I was able to run 2-3 full-day strategy retreats for other CEOs/executive teams that would cover my living expenses for the rest of the month, allowing me to focus on 15Five the rest of the time. In addition, the strategy work I was doing was related to what we were building, so it was a double win in that regard.
Related to what John Seiffer
mentioned in terms of taking a Lean Startup approach, one approach that's often helpful is to find people with the problem/challenge you're looking to solve with your software/app, and charge them to solve it manually for them. This a) helps you quickly validate whether you have identified a problem people that's a big enough issue that people are actually willing to pay for it (you have not validated this unless they have given you money -- saying they *would* give you money often isn't 100% proof), and b) lets you get paid for the customer development research you need to do anyway.
If you haven't read the book The Lean Startup, I believe there's an example in there about a startup team who were wanted to build a weekly menu planning app that would produce shopping lists based on their supermarket's in-stock inventory. To start out, they stood outside a supermarket and asked people as they left if they wanted this service -- and then did it manually for them to learn what worked and what didn't. This is obviously unscalable, but it's not meant to be done long term.
If you can't or it doesn't make sense to sell your services to manually solve the problem, even if you can do a few days of work per month of high value consulting (ideally at least tangentially related to the space you're in, but if not it may still be a good idea), you could really cut down on your burn rate and extend your runway.