Q:

How much damage will a creature take if it runs across lava while wearing a Ring of Water Walking?

How much damage will a creature take if it runs across a stream of lava while wearing a Ring of Water Walking?
It wouldn't sink into the lava, and it would get completely across so it wouldn't end its turn on top of the lava.  

A:

Damage Bounds 
On page 249 of the DMG, we have a section on Improvising Damage. On the chart are a few interesting things we can use to determine a damage amount.
For a lower bound we can use "stumbling into a fire pit" at 2d10. While yes, the creature didn't fall into the lava pool and only their feet are potentially directly exposed, lava is really hot (see below) and the proximity puts you at danger for second degree burns all over.
For our upper bound, we have a far more directly relevant 10d10 damage for "wading through lava". The creature isn't wading through lava here, they're walking across it, so it certainly wouldn't exceed 10d10.
My Ruling (feel free to disagree) 
Depending on how active/fresh the lava is I would put it between 2d10 and 6d10 damage. I would default to the middle of that range at 4d10 but the choice is entirely the DM's. 
Little bit of info about real-world lava and boots
Average liquification temperature of rocks is around 800-1200 degrees Celsius. Leather starts to burn around 200 degrees Celsius. Real life leather boots are going to melt really, really fast, even just being near the surface. Iron boots, with a melting point of 1538 degrees Celsius have a much better chance of survival, though they still are really good conductors of heat. The radiant heat is still going to hurt if they are even on the edge of the lava stream. 
Additional clarification
Since people are questioning whether ring of water walking works on lava, it would entirely depend on if lava is liquid, see definition of the ring. OP appears to treat lava as liquid in their question, and the spell waterwalk
 "grants the ability to move across any liquid surface--such as water, acid, mud, snow, quicksand, or lava--as if it were harmless solid ground", so yes, in D&D 5e, lava is liquid, and per the Improvising Damage section (DMG249), you can wade through or be submerged in it, real world lava mechanics notwithstanding

Ring of water walking
While wearing this ring, you can stand on and move across any liquid surface as if it were solid ground

A:

Lava is a liquid, but it is not harmless. It's up to a DM to determine how harmful.
Colloquially, and even scientifically most of the time, lava is a liquid. I don't think that's the real contention in this question. In the case of non-simulationist D&D 5e, it's reasonable to make and use the assumption that lava is a liquid.
That means the Ring of Water Walking(DMG, 193) would be effective for remaining on top of lava. But the ring is also very clear in what it does:

While wearing this ring, you can stand on and move across any liquid surface as if it were solid ground.

It's getting hot in here
What it doesn't say, is that it makes that solid ground "safe". So sure, you can walk on lava as if it's solid ground. But this solid ground is emanating a lot of heat. That puts this firmly in the realm of the DM to determine what the effect is.
The DMG (page 249), covers this:

In some cases, though, you need to determine damage on the fly. The Improvising Damage table gives you suggestions for when you do so.

There are some options in this table, but ultimately, it's going to be up to the DM to determine the value for 'standing on top of lava.'

A:

Like many other things in 5e, this is DM's call
The Ring of Water Walking appears to be an effect similar to the Water Walking spell, if perhaps a bit less protective.   The text of the spell indicates that you can still take damage from using it to cross lava, but it makes no indication of how much damage that is, or how much it might be reduced by, say, a good pair of boots.  It's enough of an edge case that if it's not specifically listed in either the description of the Ring itself, or of the Water Walking spell (it's not) then it's not likely to be answered anywhere.
Like many games, 5e has the idea that anything left fuzzy is up to DM's discretion.  Unlike many other (current) games, 5e actively embraced this.  There are judgement calls and fuzzy rules all over the place in 5e, with the expectation that the DM can and should come to their own conclusion on the matter.  This is one of them.  There is, in fact, an entire section in the DMG on improvising damage.  (Acknowledgement to @Ghost's answer.)
If you want to look to realism (say, for the comparison between stepping on lava and falling into a fire pit) then... it turns out it's back to DM's choice again.  There are totally vulcanologists out there who are happy to write web-pages to help aspiring vulcanologists not die, and "how close can you get to lava without severely damaging yourself" is absolutely on-topic for those sorts of webpages.  It turns out... it depends.  There are different kinds of lava, and which kind of lava it is determines what kind of shell forms over it, which determines how quickly and severely it starts damaging people who happen to be nearby.  Not only does the DM get to pick their damage level, they have a pretty wide leeway in which it's pretty much accurate.

