Can a leak in a drainage pipe cause an increase in water bill?
You don't say where you're located, but around here (Massachusetts, and I'm guessing most places are the same), we get billed for both volume of water consumed and volume of sewage emitted. That's because it costs the city money to provide water, and money to dispose of sewage. In fact, costs per volume are often (always?) substantially higher for sewage than water.

Measuring volume of sewage is much harder than measuring volume of water, so generally cities will assume some ratio of incoming water to outgoing sewage. For example, 1 cubic foot of water in might imply that 0.95 cubic foot of sewage went out; given evaporation, and people watering plants, this is pretty close to the true value.

If you irrigate your lawn a lot (or leak water into your walls), your true outgoing sewage volume will be smaller (sometimes a lot smaller) than the calculated sewer volume. However, since the town can't tell the difference, you'll get billed the (sometimes a lot larger) calculated value. Since per-volume sewage fees are generally larger (e.g. a factor of 4) than water fees, this can really boost your bill (from personal experience this can add up to thousands of dollars). To avoid this, you can get a separate water meter for your irrigation system; water usage through that meter is not presumed to increase your sewage production.

But, back to your situation. If indeed water coming into your house leaks inadvertently into the environment, then the town will still bill you as if the water were actually making it back into the sewer. However, assuming you don't change your water use, this won't change your bill; it'll just mean that the town is billing you for "sewage" that is actually damaging your house rather than being treated by the town.

TL;DR: don't let water leak into your floor!