First thing is probably to make sure your tyre is inflated properly. Being heavy does lead to what's called a pinch flat a lot more.

You can find how much air your tube should hold imprinted on the sidewall of the tyre. If you're heavy, make sure it's at maximum or the upper end at all times.

If it's possible, wider tyres might alleviate the problem. Try, if it's suitable, to aim for ones with puncture protection. They roll worse than comparable tyres, yes, but they do work.

If you hit bumps or curbs, try to take weight off the wheels a bit.

This is easier for the front wheel as it hits first, just lean back and/or pull up on your handlebars a bit to lift the tyre over the bump. If you have front suspension, you basically need to do a pumping motion or else nothing is going to happen really. For the back, this gets a bit more complicated as it's basically the first step to doing jumps. Try to take curbs slow and shift your weight accordingly, you want the wheel hitting the bump to take the least amount of force possible.

Check the inner side of the tyres and the rim for sharp bits. This is general advice, but the problem of anything putting holes in your tubes is going to be made worse by putting more weight on them.