Do the effects of a Potion of Speed or Potion of Heroism disappear if the character becomes unconscious?

Question

If a player is under the effect of a potion of speed:

When you drink this potion, you gain the effect of the haste spell for 1 minute (no concentration required). The potion’s yellow fluid is streaked with black and swirls on its own.

or a potion of heroism:

For 1 hour after drinking it, you gain 10 temporary hit points that last for 1 hour. For the same Duration, you are under the effect of the bless spell (no concentration required). This blue potion bubbles and steams as if boiling.

Do those effects wear off if the player is reduced to 0 hit points and becomes unconscious before the 1-hour duration is over?

Answer 1

They don't wear off

As you indicate, the duration of either potion is one hour, and there is no limitation should the affected creature become unconscious.

The general rules for magic items also make no mention of such a limitation. In fact, this is the only excerpt I could find addressing durations and potions:

Many items, such as potions, bypass the casting of a spell and confer the spell's effects, with their usual duration. Certain items make exceptions to these rules, changing the casting time, duration, or other parts of a spell.

In the case of a potion of speed and potion of heroism, the exception to the Spellcasting rules comes in the removal of the Concentration requirement (the usual reason a spell would end should the caster become Unconscious).

Since there is nothing that would stop the effect if the creature is unconscious, the effect will last the entire hour regardless.

Answer 2

No

The unconscious condition would end the spell effects if they required concentration. The potion's specifically state that is not the case.