This is how I handle it in my group, and why:
The reason this question comes up is because the rules for Dwarves on page 20 of the PHB assume that variant rules aren't being used. The thing is, if we ARE using the variant Encumbrance rules we have a situation that has no clear answer in the rules. As I looked around online it is one of those things that hasn't been addressed in any official publication from WOTC and so DMs have to rule on it themselves.
So my approach is to understand what the intent is by the game devs. What is their justification for the rules? What's being simulated and what are they looking to achieve?
In the PHB on page 20, the Dwarven Speed racial trait says "Your speed is not reduced by wearing heavy armor." Ok. So it seems like they're saying Dwarves are really strong, right?
Well, no. Dwarves have absolutely no particular advantage in Strength. In fact, they might be regarded as being average in Strength since other races can get Strength increases beyond the 18 maximum while Dwarves do not. (Dwarves get advantages in Constitution.) Ok. So why then would armor not slow them down if it has nothing to do with their physical Strength? Well, there's no obvious answer in the text, but we do know from the physical description of Dwarves that they're short in stature and stout... weighing about the same as a human but being around half of their height. What else do we know about Dwarves? (and yes, we're getting a little meta here) We know that they're often fine craftsmen and know how to work materials.
Because something about Dwarves has to justify a rule that allows them to ignore speed reductions from heavy armor that is not related to their physical strength. To reiterate: There's nothing at all special about Dwarven strength either in the rules or in the fluff. A Dwarf's speed (under the normal Encumbrance rules) is not reduced by wearing heavy armor whether his strength is 19, 14 or 10. So their ability to ignore the speed penalty from armor comes elsewhere.
Let's put a pin in that.
Looking at the Heavy Armor rule on page 144, it says "Heavier armor interferes with the wearer's ability to move quickly, stealthily and freely. If the Armor table shows "Str 13" or "Str 15" in the Strength column for an armor type, the armor reduces the wearer's speed by 10 feet unless the wearer has a Strength score equal to or higher than the listed score."
Ok, so things like moving stealthily or moving freely are things associated with Dexterity. I assume the intent here is to suggest that, in order to overcome the stiffness or movement burden of this armor, a certain amount of Strength is required.
... and yet, Dwarves get to ignore this rule on the grounds that they're Dwarves, not because they're strong.
Now let's look at the Variant Encumbrance rule. On Page 176 it says "When you use this variant, ignore the Strength column of the Armor table in Chapter 5." This effectively nullifies the Heavy Armor rule from page 144, and that, in turn nullifies the Dwarven racial trait benefit.
So if we go simply by that logic, the answer is
A: Dwarves lose that special ability.
But that doesn't feel right to me. We know that the Dwarven armor advantage has nothing to do with strength, and yet removing the Strength restriction from the Armor Table removes that movement advantage from them.
Now, the way I see it, this is a problem that arises from the fact that the normal Encumbrance rules are deliberately oversimplified to make things easier. The problem is that it introduces this kind of need for weird rules like Dwarves not being particularly strong and yet able to overcome the disadvantages of wearing heavy armor. I really think that the developers made a mistake in tying armor issues to Strength rather than another attribute that makes a LOT more sense in these cases:
Medieval armor was specially designed to move with the wearer. Contrary to popular misconception, High Medieval Period knights in full plate armor had no trouble at all moving around in their armor, and were even able to vault into the saddle of their horse (No, they did NOT have to be lifted into the saddle with a crane or winch. That's stupid and if you believe that you really ought to do some more historical study.) yes, armor ie heavy, but it doesn't restrict movement all that much. (This is why heavy armor had to be custom made and was extremely expensive.)
So it seems to me that D&D already has an error by taking away the Dex bonus to AC for heavy armor and limiting it for Medium armor. That said, I'm not willing to house rule away those restrictions at this time because of the potential for game balance issues, but I do think we can use this insight to solve our problem here. Therefore, here's my house ruling regarding Dwarven racial trait speed and heavy armor when using the Variant Encumbrance rule:
A: A Dwarf wearing heavy armor still gets his/her Dex bonus to AC up to +2, just as if it were Medium Armor instead.