I'd say one big disadvantage would be that you may never come out ahead financially. I have a typical gas water heater, not high efficiency, and in the summer, when it's the only gas load, my bill is less than $20. And that includes actual water usage as well as the standby losses, which obviously aren't very substantial..
You may also find that to get the right size tankless requires a larger gas service, increasing gas pipe size from the meter to where the tnakless is located, etc.
If you can, do a simple experiment sometime. When you're going to be away for a few days to a week, record the gas meter reading, make sure the water heater is the only usage, ie turn off any other pilot lights, heat, etc., then read it again when you return. That will tell you how much you are paying for standby losses.
And as others have pointed out, these losses can be cut down substantially by going to a high eff gas tank water heater, which is still typically a lot less than the install of a tankless. These close off the exhaust vent when the burner is not running, which substantially cuts the heat loss. I'd be looking at those when it's time for a new water heater. However, one other thing to consider is there is currently an energy tax credit in effect and some of these solutions could get you a credit for up to 30% of the cost. If you have a tank that is nearing the end of it's life, now may be the time to replace it with something.