Do directives to "go jump in the lake" or "off a pier" allude to drowning?

No, they are not used in a way that would suggest harm to the person to whom it is said. They hint at a frustration or impatience by the speaker that the person is deemed to have caused.

With regard to water, it would have more in common with telling someone to 'take a cold shower', to sober up (their thoughts). It's usually said to a person in response to something they have said, so harm or physical harm would be pretty innappropriate. The real message, is "get lost", "go away", and maybe also, gain a little humility by getting a little wet.

They are facetious remarks that would be said in annoyance. The person saying them doesn't literally want the other person to go and die.