[RPG] Does climbing during combat consume your Action

Drinking/administering a potion takes an action, as you cited from DMG at p.139. That's the general rule in effect, unless specifically contradicted.

You note that the DMG's description of the Potion of Healing doesn't mention that it requires an action to consume, and seem to be wondering if that omission is meant to signal something. But every other potion described in the DMG also omits any description of the mechanics of drinking potions: they just start with phrases like "when you drink..." or "for one hour after drinking...." So we can't read anything into the omission, or the general rule would apply in no cases. (DMG5e pp.187-8)

So why does the PHB call out the consume action in its Potion of Healing description?

The Potion of Healing is the only potion described in the PHB (p. 153); the PHB doesn't have a general "potions" section akin to the DMG's on p.139. So this is the natural place in the PHB to mention that interaction with the action economy¹. Would it have been clearer to add to the sidebar on p.190? Maybe, but that's a sidebar full of free interactions, not examples of things that aren't-free-interactions-though-you-might-think-they're-just-the-same.

Sidebar: But I can quaff an entire flagon...

Yeah, this bugged my table too. We made in-world sense of it by saying that a magical potion is an... unusual enough mouth-experience to take a moment² to consume. A flagon of ale... apparently the game designers assume our characters are more-inured to that experience than to the experience of drinking spells.

^{1 - it's also mentioned in the Herbalism Kit, but that's the least-natural place to talk about the potion's consumption.}

^{2 - I think about trying to get my kids to take emoxicillin and Wish it only took six seconds =P}

Only if you fail a dice roll

A character cannot "fail an ability check", because characters don't make ability checks. Ability checks are nothing but game mechanics, they are basically dice rolls. Characters take actions instead — these actions, not dice rolls, are the subject to be succeed or failed.

PHB page 6 "How to play" describes the basic gaming pattern:

1. The DM describes the environment.
2. The players describe what they want to do.
3. The DM narrates the results of the adventurers' actions. Describing the results often leads to another decision point, which brings the flow of the game right back to step 1

The DM might (but don't have to) ask the player for an ability check between steps 2 and 3. They might just say "yes, you did it" without any rolls. That depends on the action the character does (the very action the player announced at step 2).

At step 2 you describe your character's intent — the DC for an ability check will be based on this intent. If you announce "I'm trying to activate a spell scroll" it is one particular intent. If you say "I'm trying to destroy a spell scroll by activating it incorrectly" it is another intent. DM might ask for an ability check in the second case, or just allow it automatically — it isn't specified in the rules, so is completely up to the DM. The same for the improper use of Amulet of the Planes.