I got hit with an argument from one of my players the other day, and it goes like this:

He is playing a Monk and his unarmed attacks count as magical attacks. He was trying to punch a Web. I told him his bludgeoning attacks did no damage since it was immune to bludgeoning damage. My party disagreed with me, on the basis that his unarmed attacks are magical.

My question to this group is... In cases like fighting a werewolf it says 'resistance bludgeoning from non-magical attacks' but other creatures in the game it just say it is 'resistant to bludgeoning'.

Do magical weapons and attacks cancel out the resistance if it does not specifically say so?

Or

Are these creatures resistance or immune to that damage type regardless of the weapon used?

Webs are immune to Bludgeoning Damage, Magical or not

The statblock for Webs found in the Dungeon Master's Guide doesn't qualify the damage immunity as non-magical:

Each 10-foot cube of giant webs has AC 10, 15 hit points, vulnerability to fire, and immunity to bludgeoning, piercing, and psychic damage.

—Webs, Dungeon Master's Guide, pg. 105

Normally, when damage immunity is meant to be qualified against whether the damage is magical or not, the statblock will expressly say so, like with this devil I pulled out of Mordenkainen's Tome of Foes:

Damage Resistances cold; bludgeoning, piercing, and slashing from nonmagical attacks that aren't silvered

—Amnizu, Mordenkainen's Tome of Foes, pg. 164

Because this qualifier doesn't appear for webs, then the monk's damage being magical doesn't matter; they do not deal damage using their unarmed strikes.

Bear in mind that, at least for basic webs, it's possible for creatures to break out of them with a relatively easy Strength or Dexterity check. So if the monk were themselves trapped, I would encourage them to just break free with what should be a relatively good Dexterity score.

Alternately, tell them to keep a Sickle (a valid Monk weapon) on hand in the future, since they'll be allowed to use their Monk damage when using the Sickle.

It's all about the stat block

Each Monster's stat block will provide you with what is resistant/immune and what it takes to bypass that.

Those blocks are not universal and each one is specific to the creature defined within.

If something lists "resistant to bludgeoning", it is resistant to all types, including magical.

If it contains the phrase "from non-magical", then magical attacks bypass.

If it contains "silvered", then a non-magical, but silvered weapon will bypass.

The case of Web (the spell)

I'm not sure if the Web your player was targeting was from the spell, but if it was, the only listed way to destroy it by fire:

The webs are flammable. Any 5-foot cube of webs exposed to fire burns away in 1 round, dealing 2d4 fire damage to any creature that starts its turn in the fire.

It doesn't have HP, AC, or anything else. It only has a mechanic that involves it being burned.

The case of Webs (as Dungeon Hazards...DMG 105)

These webs from Giant Spiders do have a more specific stat block (my emphasis):

Each 10-foot cube of giant webs has AC10, 15 hit points, vulnerability to fire, and immunity to bludgeoning, piercing and psychic damage.

In this case, the Giant Spiders webs are immune to all bludgeoning damage from your PC - magical or not.