Can a Polearm Master Paladin cast spells while holding their polearm (e.g. a Glaive)? Considering the weapon is Heavy AND Two-handed, I think they wouldn't have a free hand to clutch the holy symbol.
They can indeed cast spells, for multiple reasons.
All page references are for the D&D basic rules v0.3 but similar information exists in the player's hand book:
Firstly, a two handed weapon is defined as follows (pg. 47, emphasis added)
This weapon requires two hands when you attack with it.
At any other time besides the actual attack with the weapon, it is possible to wield the weapon with one hand. For example, resting the weapon on the ground, or on your shoulder, while you cast spells. Thus, the paladin can rest the glaive in such a manner with one hand, and is free to use the other hand for his holy symbol.
If you want a more "word of god" confirmation, the game's designers tweeted that they would rule the two handed weapon similarly. That is:
Yeah, I would allow it too. A two-handed weapon needs two hands to be used, but not necessarily two to be carried.
There is no need to even drop the weapon as a free action.
Heavy weapons are defined as follows (pg. 46):
Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon’s size and bulk make it too large for a Small creature to use effectively.
This property has nothing to do with free hands or actions restrictions. It only applies disadvantage to attack rolls when small creatures use it. A heavy weapon doesn't even necessarily require two hands to wield, so this property alone does not restrict spellcasting.
Not all paladin spells require somatic or material components. All of the smite spells (Searing Smite, Thunderous Smite, et al.) only require verbal components, and thus as long as the paladin can speak, he can cast these spells, holy symbol or no.
Derek's answer is correct about the specific rule of the holy symbol as a focus. That excerpt can be found on page 49 of the basic rules.
A cleric or paladin can use a holy symbol as a spellcasting focus, as described in chapter 10. To use the symbol in this way, the caster must hold it in hand, wear it visibly, or bear it on a shield.
The answer is usually. Let's work through it.
Any given spell requires some combination of verbal, material, and somatic components. All spells require at least one, many two or all three. Verbal components are not an issue here.
Paladins may substitute a holy symbol as a spellcasting focus in place of any material component that doesn't have a specified cost. The description of a holy symbol includes "To use the symbol in this way, the caster must hold it in hand, wear it visibly, or bear it on a shield." Thus, an amulet need only be worn visibly in order to satisfy the material component of most spells. This is a specific exception to the general rule of needing to handle a spellcasting focus. So, if a spell has no somatic component, a paladin with a holy symbol amulet has no need of a free hand.
Any creature may interact with one item during as part of its action or movement each turn. Dropping an item, according to the developers, isn't even considered that, and can be done freely. So, if a spell has a somatic component, a paladin may freely drop the weapon from one hand, cast the spell, and pick the weapon back up into both hands using an object interaction, though even that may not warrant being called an interaction.
So, unless a paladin is not visibly wearing their holy symbol, they may easily cast any spell they have prepared while wielding a two handed weapon.