Yes, they can move across liquids and surfaces while wearing armor RAW
Unarmored Movement is a feature that has two benefits: one is a speed increase and one allows the monk to move over different surfaces.
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield.
Note how the armor/shield clause is specifically tied to the speed increase alone. It isn't even tied to a generic requirement for the whole Unarmored Movement feature, just the speed. And there appears to be no connection between the speed benefit and the surface benefit besides being under the same feature name.
I can't really see any way to interpret this other than that the requirement for not wearing armor or shield only applies to the bonus to your speed.
The only kind-of maybe hint that it might not be could be from the next phrase:
At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during the move.
The fact that it says "gain" could be interpreted as being meant to imply that this is a benefit that is on top of and inherently connected to the speed boost. I think this is a huge stretch though. I think the more reasonable reading of that word is just that it indicates that the PC is gaining that new capability.
There is no indication for what is intended here
There doesn't seem to be any explicit indication of what was actually intended here (no JC tweets or anything in the Sage Advice Compendium).
The title of the feature certainly might seem to imply something of the intent here, but that is undercut by the fact that titles of spells and features often, by the intended reading, do not reflect the ability it is naming (see catnap as a good example of a poorly named spell). Titles don't really have any mechanical effect on the ability and serve only as a label.
However, I think that the good money would be on this being an oversight. Many of a monk's features are inherently tied to the archetype envisioned for this class which is the martial character who uses simple weapons or their body as weapons and wears flowing robes instead of armor. And that would make even more sense here being that the picture of a monk stepping lightly over liquid while wearing full plate is a bit ridiculous.
But, on the other hand, not all of the features are actually tied to this so it could go either way.
I'm not seeing anything in the RAW that would indicate it is a requirement though.