There's nothing bad about wanting to save money, or do things in a more efficient way (latter being greener also).
A dishwasher can cheap/ efficient to run, we basically have like 2 days of dishes go in there, and cram it in, and we set it to come on when energy is cheap, about 2am on an eco cycle which is about 4.5 hours, so it's ready to empty when we get up, but we have a tarriff to be able to do that.
This intrigued me though, so had a daft attempt at working it out.
I think dishwashers use about 1kW of energy, per cycle, and there's not a great deal in it from the most efficient to the least efficient cycles, ours ranges from 0.8kW to about 1.4kW I think, per full cycle. So for us that's about 15p to run it, and then maybe 10p for the tablet. So, 25p for two days, which would be hard to beat, I expect.
A kettle is ~2.5kW, but only runs for what 5 minutes? So you're only using maybe a 10th of the hourly use, so 250W, but how many times would you be doing that over 2 days, just to wash up? 4-6? Probably works out a bit more than a dishwasher, maybe 50% more, around 1.5kW of electric?
A gas boiler is using ~20-40kW per hour, but that's kW of Gas, which is about 1/4 of the price of electric (tarriff dependant). If it's a combi and you're running the tap for 5 minutes then that's maybe 2-4kW of gas, plus whatever it takes to wind up, which I bet isn't efficient, never mind running the water through the pipes for such a low volume of water actually required. Do that a lot of times for any use, and it won't work out cheap or efficient. Also boilers are not top end efficiency, unless brand new and top of the range etc (which most aren't).
Long winded answer, but I expect there isn't a great deal in it either way, and it's more actually about efficiency from combining tasks, albeit not that simple in some circumstances. Nobody wants a 2 day stack of stinking (and sticking) dishes, where as having them locked away in a dishwasher is fine.
Electric heaters and electric water heaters are extremely efficient though, the trick with them is use short and sharp bursts, as the unit cost of electric is higher (in most cases).