One of my players in my dungeons and dragons fifth edition game wants to try using Phantasmal force to create an illusion that he is made of lava to stop creatures from touching him. I wasn't sure if that is something the spell is able to do even when reading through the description.
On a failed save, you create a phantasmal object, creature, or other visible phenomenon of your choice that is no larger than a 10-foot cube and that is perceivable only to the target for the duration. (Emphasis mine)
A body-shaped mass of lava definitely counts as a 'visible phenomenon' (which is almost an all-encompassing term anyway). The character can certainly fit inside a '10-foot cube', so that requirement is met. If you were unsure as to whether the illusion could occupy space already taken up by the caster themselves, then they could simply create an illusion of a lava body covering, just outside the skin, bypassing that problem.
The player could even argue that the d6 damage per round could be perceived as lava flying from their body, and onto the creature, although you may want to disallow that.
Yes, they can, against one target only. As seen in the spell description (PHB p264):
While a target is affected by the spell, the target treats the phantasm as if it were real. The target rationalizes any illogical outcomes from interacting with the phantasm. For example. a target attempting to walk across a phantasmal bridge that spans a chasm falls once it steps onto the bridge. If the target survives the fall, it still believes that the bridge exists and comes up with some other explanation for its fall-it was pushed, it slipped, or a strong wind might have knocked it off.
Similarly, a phantasm created to appear as fire, a pool of acid, or lava can burn the target. Each round on your turn. the phantasm can deal 1d6 psychic damage to the target if it is in the phantasm's area or within 5 feet of the phantasm, provided that the illusion is of a creature or hazard that could logically deal damage, such as by attacking. The target perceives the damage as a type appropriate to the illusion.