Wall of Force does not transfer thermal energy.

The Player's Handbook describes the spell as:

immune to all damage

Cold and fire are both types of damage, and in your example we can assume that the lava would cause fire damage.

The question, then is "Can Wall of Force allow damage to pass through it if it does not take damage itself?" The answer still is no, and here's why:

There are three types of heat transfer: Conduction (contact between two objects, such as in the spell Heat Metal), Convection (which happens primarily in gases and liquids, of which the Wall of Force is neither), and Radiation (does not require a medium, and consists of electromagnetic waves).

We can rule out Convection and Conduction, since the Player's Handbook describes Force as:

pure magical energy

which more than likely does not behave the same as normal matter. In addition, if Force did act in a similar way, then we have a precedent for an amount of Fire damage that does not pass through a Wall of Force, namely a Fireball spell cast at 9th level. This will do the base 8d6 fire damage, plus an additional 5d6 fire damage for a maximum of 78 fire damage. Given that there is no limit to the number of creatures that can take a turn in a round, we can assume that even with even three spellcasters casting 9th level Fireball spells, the wall (and the creature being protected by the wall) will remain unharmed.

Per the Improvising Damage chart on page 249 of the Dungeon Master's Guide,

18d10 Being submerged in lava, ...

That's a maximum of 180 fire damage per round submerged in lava. With 3 9th-level Fireball spells maxing out at 234 fire damage per round, the Wall of Force has precedent for not allowing extreme temperatures to affect the enclosed creature.

As far as heating the interior by Radiation,

Nothing can physically pass through the wall.

That would include things on the electromagnetic scale, since the waves would physically need to pass through the wall, and light would be an exception (not the rule), since the wall is specified to be invisible.

The Dragon Masks from the Rise of Tiamat Supplement

The Rise of Tiamat Supplement contains notes on several legendary magic Dragon Masks, one for each color of chromatic dragon. All masks share the same relevant property:

Damage Absorption. You have resistance against the mask’s damage type. If you already have resistance to that damage type from another source, you instead have immunity to that damage type. If you already have immunity to that damage type from another source, whenever you are subjected to damage of that type, you take none of that damage and regain a number of hit points equal to half the damage dealt of that type.

If you somehow gain immunity to fire damage and attune to the Red Dragon mask (which is associated with fire), you will take no fire damage and instead be healed for half of the fire damage that would have been dealt.