I am a 44 year old man that is very heavy (250–260lbs(113-118kg)) and I am looking to buy my first mountain bike to do some riding on the weekend to burn off some of this fat I have.
250-260 pounds isn't that heavy.
You may not need to worry about weight at all.
See if you can find the information from the manufacturer about weight limits for their bikes. For a mountain bike that you intend to use for what it was made for, you're probably within the weight limit the bike is designed for.
You may want to avoid the lightest weight stuff (carbon) unless you check the weight specs. But that's mostly stuff on the more expensive bikes.
Some of the parts that normally wear out will likely wear out faster for you than for somebody lighter, but that's no big deal.
If you plan to do tougher things with the bike than it was made for (jump a trail bike, for instance) you could run into problems. Stick with getting a bike that's intended for what you plan to do with the bike.
2. Specialized Bike Warranty Policy (PDF) (weight info near the bottom):
In other words: avoid the carbon/composite seat posts, handlebar stems and handlebars and you'll be fine with this brand.
3. Cannondale 2010 User's manual (PDF) (Starting at page 52)
Don't go full suspension - it at $1000 price point its a gimmick, and at your weight, you would need highend stuff for it to be adjustable enough to be useful. I am sure you can spend less, (I ride off-road with a guy who's 220lbs, hes really fit and all muscle, rides hard and fast. His sub $1000 bike stays in one piece), unless you are looking for serious off road use.
I would strongly recommend buying form your local bike shop rather than Kmart. They should be able to put you oto a bike that will meet you budget, fit you properly and give you advice. They will also be able to maintain it for you. If they can't put you on a bike within your budget look for one that has lots of kids bikes, they will understand the budget, and wont try to tell you that you need the lycra clothing to go with it.
I agree with the others: your weight is not that much of an issue for a decent quality well under $1k bike. I'm 175lbs now, but when I first started riding at 360lbs I rode a Specialized Big Hit Mountain bike. About $1,600 and overkill for the on-road riding I did.
I now use a simple Trek Hybrid, and love it.. no clunky clumsy mountain bike.. (unless of course you are using it off road.. I never did and never intended to).
The bottom line is don't overthink it. Find the weight limits. Some kind person listed Trek's above.. and there are other brands as well.. and go for it.. Just move and watch the carbs and calories and the weight will fall off.
You might look into something like the Kona Hoss, which I believe is no longer made but can probably be found used. Here are some reviews on MTBR.
The most critical thing to selecting a bike will be getting strong wheels, crankset, seat & post, pedals, and fork. These are the components that bear most of your weight. You can probably start with almost any mid-range mountain bike and just make a few changes:
Bottom line: You don't need to spend $1000 plus, unless you want to. You can probably find a decent used bike on Craigslist for $200-300, and then budget another $200 for some simple upgrades. You should also invest in a nice pair of bike shorts. There are lots of non-spandex options that look just like regular street shorts. Once you start riding and losing weight you may find you want a different type of bike or something fancier...
As someone who weighs in at 375lbs, I can tell you that a good build will hold up just fine. I would stay away from box-store bikes (Walmart, K-Mart, etc) and shady bike shops, as they tend to cut corners in components and overall build quality. Hit up your local bike shop and let them know your worries. A good bike shop will probably have some great suggestions for you.
In my case, I prefer steel bikes (started on a Trek, then went to Surly). I know that From my understanding, aluminum bikes are stronger, but I have a mental block against "lighter", so I stick with steel. The only problem I have run into was wheel quality. Once I went to hand-built wheels with quality components, I stopped having problems. Right now I ride a Surly Ogre with 36 spoke wheels atop Surly New Disc hubs; they are pretty much bullet-proof.